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Properties

Label 10-3330e5-1.1-c1e5-0-1
Degree $10$
Conductor $4.095\times 10^{17}$
Sign $1$
Analytic cond. $1.32925\times 10^{7}$
Root an. cond. $5.15656$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 15·4-s + 5·5-s + 3·7-s + 35·8-s + 25·10-s + 5·11-s + 5·13-s + 15·14-s + 70·16-s + 7·17-s − 19-s + 75·20-s + 25·22-s + 5·23-s + 15·25-s + 25·26-s + 45·28-s + 2·29-s + 16·31-s + 126·32-s + 35·34-s + 15·35-s + 5·37-s − 5·38-s + 175·40-s + 2·41-s + ⋯
L(s)  = 1  + 3.53·2-s + 15/2·4-s + 2.23·5-s + 1.13·7-s + 12.3·8-s + 7.90·10-s + 1.50·11-s + 1.38·13-s + 4.00·14-s + 35/2·16-s + 1.69·17-s − 0.229·19-s + 16.7·20-s + 5.33·22-s + 1.04·23-s + 3·25-s + 4.90·26-s + 8.50·28-s + 0.371·29-s + 2.87·31-s + 22.2·32-s + 6.00·34-s + 2.53·35-s + 0.821·37-s − 0.811·38-s + 27.6·40-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{10} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{10} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 3^{10} \cdot 5^{5} \cdot 37^{5}\)
Sign: $1$
Analytic conductor: \(1.32925\times 10^{7}\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{5} \cdot 3^{10} \cdot 5^{5} \cdot 37^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(577.2522934\)
\(L(\frac12)\) \(\approx\) \(577.2522934\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{5} \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{5} \)
37$C_1$ \( ( 1 - T )^{5} \)
good7$S_5\times C_2$ \( 1 - 3 T + 2 p T^{2} - 29 T^{3} + 97 T^{4} - 160 T^{5} + 97 p T^{6} - 29 p^{2} T^{7} + 2 p^{4} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 5 T + 26 T^{2} - 109 T^{3} + 463 T^{4} - 1776 T^{5} + 463 p T^{6} - 109 p^{2} T^{7} + 26 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 5 T + 33 T^{2} - 148 T^{3} + 746 T^{4} - 2526 T^{5} + 746 p T^{6} - 148 p^{2} T^{7} + 33 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 7 T + 80 T^{2} - 413 T^{3} + 155 p T^{4} - 10080 T^{5} + 155 p^{2} T^{6} - 413 p^{2} T^{7} + 80 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + T + 9 T^{2} + 116 T^{3} + 620 T^{4} + 54 T^{5} + 620 p T^{6} + 116 p^{2} T^{7} + 9 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 5 T + 83 T^{2} - 340 T^{3} + 3370 T^{4} - 10734 T^{5} + 3370 p T^{6} - 340 p^{2} T^{7} + 83 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 2 T + 110 T^{2} - 208 T^{3} + 5617 T^{4} - 8796 T^{5} + 5617 p T^{6} - 208 p^{2} T^{7} + 110 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 16 T + 192 T^{2} - 1512 T^{3} + 10431 T^{4} - 59424 T^{5} + 10431 p T^{6} - 1512 p^{2} T^{7} + 192 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 2 T + 110 T^{2} - 154 T^{3} + 6757 T^{4} - 5520 T^{5} + 6757 p T^{6} - 154 p^{2} T^{7} + 110 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 12 T + 176 T^{2} - 1448 T^{3} + 13399 T^{4} - 81640 T^{5} + 13399 p T^{6} - 1448 p^{2} T^{7} + 176 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 6 T + 115 T^{2} - 624 T^{3} + 8194 T^{4} - 44244 T^{5} + 8194 p T^{6} - 624 p^{2} T^{7} + 115 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 3 T + 214 T^{2} - 495 T^{3} + 20305 T^{4} - 35472 T^{5} + 20305 p T^{6} - 495 p^{2} T^{7} + 214 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 10 T + 179 T^{2} + 1472 T^{3} + 17638 T^{4} + 122892 T^{5} + 17638 p T^{6} + 1472 p^{2} T^{7} + 179 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 16 T + 138 T^{2} - 24 p T^{3} + 15201 T^{4} - 121128 T^{5} + 15201 p T^{6} - 24 p^{3} T^{7} + 138 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 4 T + 87 T^{2} + 528 T^{3} + 162 T^{4} + 69288 T^{5} + 162 p T^{6} + 528 p^{2} T^{7} + 87 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 8 T + 203 T^{2} + 1792 T^{3} + 23794 T^{4} + 169200 T^{5} + 23794 p T^{6} + 1792 p^{2} T^{7} + 203 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 3 T + 113 T^{2} + 688 T^{3} + 12934 T^{4} + 46682 T^{5} + 12934 p T^{6} + 688 p^{2} T^{7} + 113 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 2 T + 219 T^{2} - 88 T^{3} + 26378 T^{4} - 5676 T^{5} + 26378 p T^{6} - 88 p^{2} T^{7} + 219 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 5 T + 239 T^{2} + 2176 T^{3} + 25930 T^{4} + 290022 T^{5} + 25930 p T^{6} + 2176 p^{2} T^{7} + 239 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 7 T + 149 T^{2} + 1148 T^{3} + 17026 T^{4} + 177306 T^{5} + 17026 p T^{6} + 1148 p^{2} T^{7} + 149 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 14 T + 438 T^{2} - 5026 T^{3} + 81785 T^{4} - 710808 T^{5} + 81785 p T^{6} - 5026 p^{2} T^{7} + 438 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.03452049055897968325337661823, −4.99067786737013644893601887204, −4.80061832045932372725559662541, −4.71959102058278242584747052815, −4.40210054818457493706053114869, −4.30808840965176781475774275941, −4.22515266862683460625669271459, −3.96023091781569812895884972124, −3.88941798357273735054150974196, −3.81223998162089327880322477435, −3.21082279885002450848933891192, −3.18238610107838541369848184101, −3.15877888806595607379543982229, −2.97526942731867119785507463673, −2.93353842991766621689934137868, −2.44814425585896048930413788408, −2.25390352748733261718103420577, −2.15604879615561611124697767487, −2.11501647086204612004499910823, −1.80431656127142705863334910735, −1.39701271638754394976802518131, −1.17974416425608377846294685597, −1.12084262714109517924855801461, −1.00195523131641969999236905200, −0.881136298997573516803859375059, 0.881136298997573516803859375059, 1.00195523131641969999236905200, 1.12084262714109517924855801461, 1.17974416425608377846294685597, 1.39701271638754394976802518131, 1.80431656127142705863334910735, 2.11501647086204612004499910823, 2.15604879615561611124697767487, 2.25390352748733261718103420577, 2.44814425585896048930413788408, 2.93353842991766621689934137868, 2.97526942731867119785507463673, 3.15877888806595607379543982229, 3.18238610107838541369848184101, 3.21082279885002450848933891192, 3.81223998162089327880322477435, 3.88941798357273735054150974196, 3.96023091781569812895884972124, 4.22515266862683460625669271459, 4.30808840965176781475774275941, 4.40210054818457493706053114869, 4.71959102058278242584747052815, 4.80061832045932372725559662541, 4.99067786737013644893601887204, 5.03452049055897968325337661823

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