Properties

Degree $4$
Conductor $2160900$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 5-s + 6-s − 8-s − 10-s − 4·13-s − 15-s − 16-s + 6·17-s − 4·19-s − 24-s − 4·26-s − 27-s − 12·29-s − 30-s + 8·31-s + 6·34-s − 2·37-s − 4·38-s − 4·39-s + 40-s + 12·41-s − 8·43-s − 48-s + 6·51-s + 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 0.447·5-s + 0.408·6-s − 0.353·8-s − 0.316·10-s − 1.10·13-s − 0.258·15-s − 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.204·24-s − 0.784·26-s − 0.192·27-s − 2.22·29-s − 0.182·30-s + 1.43·31-s + 1.02·34-s − 0.328·37-s − 0.648·38-s − 0.640·39-s + 0.158·40-s + 1.87·41-s − 1.21·43-s − 0.144·48-s + 0.840·51-s + 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2160900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2160900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.300352486\)
\(L(\frac12)\) \(\approx\) \(2.300352486\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.999455828235689195457185090904, −9.257930923204582788890108078237, −8.957708445990459306798124837534, −8.345863822808380456985608810940, −8.202916362463630845185988018292, −7.58932423213480981116045275959, −7.34164395307515912174007795081, −7.08461696689027482813478522335, −6.20537445365979675710518545164, −6.08720726301409836765464791644, −5.42834896561593495817217369094, −5.19853337791433568670991538822, −4.40953861238664577893831957836, −4.35974068274400443492239123835, −3.65863340018501041038093033770, −3.28978818186339699449988687586, −2.77959941454233904484367971001, −2.24408576449534734034468652085, −1.59253655493677423458052276415, −0.51118839732914461250351631195, 0.51118839732914461250351631195, 1.59253655493677423458052276415, 2.24408576449534734034468652085, 2.77959941454233904484367971001, 3.28978818186339699449988687586, 3.65863340018501041038093033770, 4.35974068274400443492239123835, 4.40953861238664577893831957836, 5.19853337791433568670991538822, 5.42834896561593495817217369094, 6.08720726301409836765464791644, 6.20537445365979675710518545164, 7.08461696689027482813478522335, 7.34164395307515912174007795081, 7.58932423213480981116045275959, 8.202916362463630845185988018292, 8.345863822808380456985608810940, 8.957708445990459306798124837534, 9.257930923204582788890108078237, 9.999455828235689195457185090904

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