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Properties

Label 4-156e2-1.1-c1e2-0-13
Degree $4$
Conductor $24336$
Sign $1$
Analytic cond. $1.55168$
Root an. cond. $1.11609$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 6·7-s + 6·11-s − 5·13-s − 3·17-s − 6·19-s + 6·21-s − 6·23-s + 7·25-s − 27-s − 9·29-s + 6·33-s − 9·37-s − 5·39-s − 15·41-s − 2·43-s + 17·49-s − 3·51-s − 18·53-s − 6·57-s + 24·59-s + 11·61-s + 18·67-s − 6·69-s + 18·71-s + 7·75-s + 36·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 2.26·7-s + 1.80·11-s − 1.38·13-s − 0.727·17-s − 1.37·19-s + 1.30·21-s − 1.25·23-s + 7/5·25-s − 0.192·27-s − 1.67·29-s + 1.04·33-s − 1.47·37-s − 0.800·39-s − 2.34·41-s − 0.304·43-s + 17/7·49-s − 0.420·51-s − 2.47·53-s − 0.794·57-s + 3.12·59-s + 1.40·61-s + 2.19·67-s − 0.722·69-s + 2.13·71-s + 0.808·75-s + 4.10·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24336\)    =    \(2^{4} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1.55168\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 24336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.712976790\)
\(L(\frac12)\) \(\approx\) \(1.712976790\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.5.a_ah
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.7.ag_t
11$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_x
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.g_bf
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_ca
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.j_cm
41$C_2^2$ \( 1 + 15 T + 116 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.41.p_em
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_abn
47$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.47.a_ade
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2^2$ \( 1 - 24 T + 251 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.59.ay_jr
61$C_2^2$ \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.61.al_ci
67$C_2^2$ \( 1 - 18 T + 175 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.67.as_gt
71$C_2^2$ \( 1 - 18 T + 179 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.71.as_gx
73$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \) 2.73.a_aep
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \) 2.83.a_afy
89$C_2^2$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_fh
97$C_2^2$ \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.97.am_fp
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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

−12.92969730931887237355674837410, −12.91720984623017817836612683453, −11.86023146607430678276249578473, −11.84788915870168825750305157953, −11.15827558278724835044081022692, −10.96526636825649627149395754262, −9.962720711660750300781971406164, −9.761667962638959940247021542591, −8.711229340482488262986780144509, −8.623581491597419451792796482663, −8.249799728223798369499347931976, −7.44773090777286749523056087306, −6.86948085074299929749018448082, −6.45984664369464484242152134746, −5.16726744818237835871522606195, −5.02783433508979592011012781687, −4.16420577549207164277272409888, −3.64827924370899148887378178731, −2.04421677189285670853746386208, −1.87434699414572112069597478785, 1.87434699414572112069597478785, 2.04421677189285670853746386208, 3.64827924370899148887378178731, 4.16420577549207164277272409888, 5.02783433508979592011012781687, 5.16726744818237835871522606195, 6.45984664369464484242152134746, 6.86948085074299929749018448082, 7.44773090777286749523056087306, 8.249799728223798369499347931976, 8.623581491597419451792796482663, 8.711229340482488262986780144509, 9.761667962638959940247021542591, 9.962720711660750300781971406164, 10.96526636825649627149395754262, 11.15827558278724835044081022692, 11.84788915870168825750305157953, 11.86023146607430678276249578473, 12.91720984623017817836612683453, 12.92969730931887237355674837410

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