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Properties

Label 10-2001e5-1.1-c1e5-0-0
Degree $10$
Conductor $3.208\times 10^{16}$
Sign $-1$
Analytic cond. $1.04140\times 10^{6}$
Root an. cond. $3.99725$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 5·3-s + 4-s − 3·5-s + 10·6-s − 5·7-s + 8-s + 15·9-s + 6·10-s − 8·11-s − 5·12-s + 5·13-s + 10·14-s + 15·15-s − 8·16-s + 2·17-s − 30·18-s − 9·19-s − 3·20-s + 25·21-s + 16·22-s + 5·23-s − 5·24-s − 7·25-s − 10·26-s − 35·27-s − 5·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.88·3-s + 1/2·4-s − 1.34·5-s + 4.08·6-s − 1.88·7-s + 0.353·8-s + 5·9-s + 1.89·10-s − 2.41·11-s − 1.44·12-s + 1.38·13-s + 2.67·14-s + 3.87·15-s − 2·16-s + 0.485·17-s − 7.07·18-s − 2.06·19-s − 0.670·20-s + 5.45·21-s + 3.41·22-s + 1.04·23-s − 1.02·24-s − 7/5·25-s − 1.96·26-s − 6.73·27-s − 0.944·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 23^{5} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 23^{5} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 23^{5} \cdot 29^{5}\)
Sign: $-1$
Analytic conductor: \(1.04140\times 10^{6}\)
Root analytic conductor: \(3.99725\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 3^{5} \cdot 23^{5} \cdot 29^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + T )^{5} \)
23$C_1$ \( ( 1 - T )^{5} \)
29$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + p T + 3 T^{2} + 3 T^{3} + 9 T^{4} + 15 T^{5} + 9 p T^{6} + 3 p^{2} T^{7} + 3 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 3 T + 16 T^{2} + 32 T^{3} + 132 T^{4} + 227 T^{5} + 132 p T^{6} + 32 p^{2} T^{7} + 16 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 5 T + 33 T^{2} + 115 T^{3} + 447 T^{4} + 1151 T^{5} + 447 p T^{6} + 115 p^{2} T^{7} + 33 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 8 T + 71 T^{2} + 359 T^{3} + 1734 T^{4} + 5961 T^{5} + 1734 p T^{6} + 359 p^{2} T^{7} + 71 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 5 T + 61 T^{2} - 237 T^{3} + 1563 T^{4} - 4463 T^{5} + 1563 p T^{6} - 237 p^{2} T^{7} + 61 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 2 T + 80 T^{2} - 125 T^{3} + 2634 T^{4} - 3097 T^{5} + 2634 p T^{6} - 125 p^{2} T^{7} + 80 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 9 T + 89 T^{2} + 438 T^{3} + 2569 T^{4} + 9583 T^{5} + 2569 p T^{6} + 438 p^{2} T^{7} + 89 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 6 T + 115 T^{2} + 685 T^{3} + 6096 T^{4} + 31077 T^{5} + 6096 p T^{6} + 685 p^{2} T^{7} + 115 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 10 T + 161 T^{2} - 1277 T^{3} + 11230 T^{4} - 67939 T^{5} + 11230 p T^{6} - 1277 p^{2} T^{7} + 161 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 11 T + 184 T^{2} + 1340 T^{3} + 12920 T^{4} + 71977 T^{5} + 12920 p T^{6} + 1340 p^{2} T^{7} + 184 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 9 T + 140 T^{2} + 922 T^{3} + 10096 T^{4} + 53457 T^{5} + 10096 p T^{6} + 922 p^{2} T^{7} + 140 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 13 T + 90 T^{2} + 567 T^{3} + 5553 T^{4} + 45003 T^{5} + 5553 p T^{6} + 567 p^{2} T^{7} + 90 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - T + 216 T^{2} - 192 T^{3} + 20850 T^{4} - 14373 T^{5} + 20850 p T^{6} - 192 p^{2} T^{7} + 216 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 6 T + 91 T^{2} - 8 T^{3} - 4194 T^{4} + 42652 T^{5} - 4194 p T^{6} - 8 p^{2} T^{7} + 91 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 23 T + 389 T^{2} - 4749 T^{3} + 48365 T^{4} - 408493 T^{5} + 48365 p T^{6} - 4749 p^{2} T^{7} + 389 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 10 T + 202 T^{2} + 1197 T^{3} + 16424 T^{4} + 70105 T^{5} + 16424 p T^{6} + 1197 p^{2} T^{7} + 202 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 11 T + 357 T^{2} + 3005 T^{3} + 50921 T^{4} + 315865 T^{5} + 50921 p T^{6} + 3005 p^{2} T^{7} + 357 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 31 T + 648 T^{2} - 9721 T^{3} + 114923 T^{4} - 1087551 T^{5} + 114923 p T^{6} - 9721 p^{2} T^{7} + 648 p^{3} T^{8} - 31 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 8 T + 333 T^{2} - 2069 T^{3} + 48048 T^{4} - 229967 T^{5} + 48048 p T^{6} - 2069 p^{2} T^{7} + 333 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 7 T + 130 T^{2} - 1342 T^{3} + 15890 T^{4} - 85583 T^{5} + 15890 p T^{6} - 1342 p^{2} T^{7} + 130 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 3 T + 295 T^{2} - 1825 T^{3} + 37881 T^{4} - 278011 T^{5} + 37881 p T^{6} - 1825 p^{2} T^{7} + 295 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 3 T + 217 T^{2} + 949 T^{3} + 10747 T^{4} + 244897 T^{5} + 10747 p T^{6} + 949 p^{2} T^{7} + 217 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.12633247855376382650470621356, −5.67937985869290355351448833926, −5.57411983379589599871664285137, −5.38115552431414949648989777807, −5.37243526086060586329972576392, −4.88901303105669850909543833790, −4.88564914494715297849902505642, −4.83853608035421550805997602634, −4.65866204721430688681813672214, −4.62922247106009666986247257325, −3.93202900147511801802400254663, −3.82280497253582013435931738854, −3.80879045957945628773915433500, −3.80454588052404640328422686168, −3.61509556081520773657488788115, −3.26971223588676075656076109747, −3.02339471892176435998969504200, −2.51223007709982505580447404393, −2.45111046656329092446787127668, −2.43042810073001041573317296685, −2.09172333698771226243855232810, −1.66180895046833995294914728141, −1.34312554582318347956188047667, −1.06224578939487790039405811184, −0.994633543089141866606701045434, 0, 0, 0, 0, 0, 0.994633543089141866606701045434, 1.06224578939487790039405811184, 1.34312554582318347956188047667, 1.66180895046833995294914728141, 2.09172333698771226243855232810, 2.43042810073001041573317296685, 2.45111046656329092446787127668, 2.51223007709982505580447404393, 3.02339471892176435998969504200, 3.26971223588676075656076109747, 3.61509556081520773657488788115, 3.80454588052404640328422686168, 3.80879045957945628773915433500, 3.82280497253582013435931738854, 3.93202900147511801802400254663, 4.62922247106009666986247257325, 4.65866204721430688681813672214, 4.83853608035421550805997602634, 4.88564914494715297849902505642, 4.88901303105669850909543833790, 5.37243526086060586329972576392, 5.38115552431414949648989777807, 5.57411983379589599871664285137, 5.67937985869290355351448833926, 6.12633247855376382650470621356

Graph of the $Z$-function along the critical line