Properties

Label 4-9680e2-1.1-c1e2-0-9
Degree $4$
Conductor $93702400$
Sign $1$
Analytic cond. $5974.54$
Root an. cond. $8.79176$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s + 2·9-s + 8·13-s − 8·17-s + 3·25-s − 4·29-s + 8·35-s − 4·37-s − 12·41-s − 12·43-s − 4·45-s − 2·49-s + 12·53-s + 8·59-s − 4·61-s − 8·63-s − 16·65-s − 8·67-s + 8·73-s + 8·79-s − 5·81-s − 12·83-s + 16·85-s − 4·89-s − 32·91-s − 4·97-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s + 2/3·9-s + 2.21·13-s − 1.94·17-s + 3/5·25-s − 0.742·29-s + 1.35·35-s − 0.657·37-s − 1.87·41-s − 1.82·43-s − 0.596·45-s − 2/7·49-s + 1.64·53-s + 1.04·59-s − 0.512·61-s − 1.00·63-s − 1.98·65-s − 0.977·67-s + 0.936·73-s + 0.900·79-s − 5/9·81-s − 1.31·83-s + 1.73·85-s − 0.423·89-s − 3.35·91-s − 0.406·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(93702400\)    =    \(2^{8} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5974.54\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 93702400,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−7.27424107664339009375534040876, −7.04341834910921791003072383394, −6.92651807524227738868224020017, −6.50050009911494966545163044702, −6.24116275827303572947426936530, −6.02639384366262849870786349163, −5.47417035202952314175710546325, −5.06145366466741594609330374741, −4.64537078367246223584656947369, −4.31135154274888583666124147644, −3.83248280817668773547373861478, −3.62849220187349808216191674271, −3.37058361941314786795676896138, −3.06618025614208884707374095689, −2.38127189102190855728440602379, −1.89553221521314065002987745726, −1.48813123601387230128999333349, −0.909460562232389895381743687476, 0, 0, 0.909460562232389895381743687476, 1.48813123601387230128999333349, 1.89553221521314065002987745726, 2.38127189102190855728440602379, 3.06618025614208884707374095689, 3.37058361941314786795676896138, 3.62849220187349808216191674271, 3.83248280817668773547373861478, 4.31135154274888583666124147644, 4.64537078367246223584656947369, 5.06145366466741594609330374741, 5.47417035202952314175710546325, 6.02639384366262849870786349163, 6.24116275827303572947426936530, 6.50050009911494966545163044702, 6.92651807524227738868224020017, 7.04341834910921791003072383394, 7.27424107664339009375534040876

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