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Properties

Label 4-364e2-1.1-c1e2-0-14
Degree $4$
Conductor $132496$
Sign $1$
Analytic cond. $8.44805$
Root an. cond. $1.70486$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 6·5-s − 2·9-s + 2·13-s + 4·16-s − 12·17-s + 12·20-s + 17·25-s − 18·29-s + 4·36-s + 4·37-s − 12·41-s + 12·45-s + 49-s − 4·52-s − 18·53-s − 20·61-s − 8·64-s − 12·65-s + 24·68-s + 22·73-s − 24·80-s − 5·81-s + 72·85-s + 30·89-s − 2·97-s − 34·100-s + ⋯
L(s)  = 1  − 4-s − 2.68·5-s − 2/3·9-s + 0.554·13-s + 16-s − 2.91·17-s + 2.68·20-s + 17/5·25-s − 3.34·29-s + 2/3·36-s + 0.657·37-s − 1.87·41-s + 1.78·45-s + 1/7·49-s − 0.554·52-s − 2.47·53-s − 2.56·61-s − 64-s − 1.48·65-s + 2.91·68-s + 2.57·73-s − 2.68·80-s − 5/9·81-s + 7.80·85-s + 3.17·89-s − 0.203·97-s − 3.39·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(132496\)    =    \(2^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(8.44805\)
Root analytic conductor: \(1.70486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 132496,\ (\ :1/2, 1/2),\ 1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.a_al
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.29.s_fj
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.43.a_dh
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.47.a_dh
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.a_gb
83$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.83.a_gb
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.89.abe_pn
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.97.c_hn
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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

−8.965474064429713311069674592028, −8.222490883796533571515292244064, −8.023034081311283559976527256533, −7.67278082338318419108930940202, −7.02868293890643359091621042595, −6.48432181419363077598714474936, −5.84786552632357662072719983468, −4.95454046107923865998450494116, −4.61390554680519199978963802626, −4.04609643315305459582542236808, −3.64885443653876343577018155811, −3.23023689233311204016664652811, −1.94321857646898970494164375533, 0, 0, 1.94321857646898970494164375533, 3.23023689233311204016664652811, 3.64885443653876343577018155811, 4.04609643315305459582542236808, 4.61390554680519199978963802626, 4.95454046107923865998450494116, 5.84786552632357662072719983468, 6.48432181419363077598714474936, 7.02868293890643359091621042595, 7.67278082338318419108930940202, 8.023034081311283559976527256533, 8.222490883796533571515292244064, 8.965474064429713311069674592028

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