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Properties

Label 4-5760e2-1.1-c1e2-0-30
Degree $4$
Conductor $33177600$
Sign $1$
Analytic cond. $2115.43$
Root an. cond. $6.78187$
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 − 2·7-s + 4·11-s + 2·13-s − 6·17-s − 2·19-s − 2·23-s + 3·25-s + 6·31-s + 4·35-s − 2·37-s + 4·41-s − 4·43-s − 18·47-s + 6·49-s − 20·53-s − 8·55-s + 12·59-s − 4·61-s − 4·65-s − 4·67-s − 16·71-s + 8·73-s − 8·77-s + 18·79-s − 8·83-s + 12·85-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.755·7-s + 1.20·11-s + 0.554·13-s − 1.45·17-s − 0.458·19-s − 0.417·23-s + 3/5·25-s + 1.07·31-s + 0.676·35-s − 0.328·37-s + 0.624·41-s − 0.609·43-s − 2.62·47-s + 6/7·49-s − 2.74·53-s − 1.07·55-s + 1.56·59-s − 0.512·61-s − 0.496·65-s − 0.488·67-s − 1.89·71-s + 0.936·73-s − 0.911·77-s + 2.02·79-s − 0.878·83-s + 1.30·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33177600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33177600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(33177600\)    =    \(2^{14} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2115.43\)
Root analytic conductor: \(6.78187\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 33177600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ac
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_4$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_k
17$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_ba
19$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_w
23$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_be
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$D_{4}$ \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_cc
37$D_{4}$ \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_cg
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_w
47$D_{4}$ \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.47.s_gc
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_cs
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_dq
79$D_{4}$ \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.79.as_io
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−7.84637329220254311161997108285, −7.82053474897148252502712547619, −7.07957666572969796562710646184, −6.74293134005035786299116079396, −6.52826779079985830934492029231, −6.33997170014930187593866680475, −6.03666693781985005113401206069, −5.40134541887683139517139156248, −4.80289290148765405220033235054, −4.67973377962354191481008131615, −4.23091385627799423712374069361, −3.85068993340626790169456639767, −3.41248756236720843537475332214, −3.28776367237713647586937669580, −2.53633362942484135456964377879, −2.21797576556738060684948907753, −1.43534406510263568371472682274, −1.15487096254857086997789610179, 0, 0, 1.15487096254857086997789610179, 1.43534406510263568371472682274, 2.21797576556738060684948907753, 2.53633362942484135456964377879, 3.28776367237713647586937669580, 3.41248756236720843537475332214, 3.85068993340626790169456639767, 4.23091385627799423712374069361, 4.67973377962354191481008131615, 4.80289290148765405220033235054, 5.40134541887683139517139156248, 6.03666693781985005113401206069, 6.33997170014930187593866680475, 6.52826779079985830934492029231, 6.74293134005035786299116079396, 7.07957666572969796562710646184, 7.82053474897148252502712547619, 7.84637329220254311161997108285

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