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Properties

Label 4-2664e2-1.1-c1e2-0-4
Degree $4$
Conductor $7096896$
Sign $1$
Analytic cond. $452.504$
Root an. cond. $4.61217$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 8·7-s − 4·11-s + 6·17-s − 2·23-s − 2·25-s + 6·29-s + 12·31-s − 16·35-s + 2·37-s + 4·41-s + 4·43-s − 16·47-s + 34·49-s − 8·53-s + 8·55-s − 2·59-s − 12·61-s + 8·67-s − 8·71-s + 8·73-s − 32·77-s + 20·83-s − 12·85-s − 22·89-s + 20·101-s − 24·109-s + ⋯
L(s)  = 1  − 0.894·5-s + 3.02·7-s − 1.20·11-s + 1.45·17-s − 0.417·23-s − 2/5·25-s + 1.11·29-s + 2.15·31-s − 2.70·35-s + 0.328·37-s + 0.624·41-s + 0.609·43-s − 2.33·47-s + 34/7·49-s − 1.09·53-s + 1.07·55-s − 0.260·59-s − 1.53·61-s + 0.977·67-s − 0.949·71-s + 0.936·73-s − 3.64·77-s + 2.19·83-s − 1.30·85-s − 2.33·89-s + 1.99·101-s − 2.29·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7096896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7096896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7096896\)    =    \(2^{6} \cdot 3^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(452.504\)
Root analytic conductor: \(4.61217\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7096896,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.296929473\)
\(L(\frac12)\) \(\approx\) \(3.296929473\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_g
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
11$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_g
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bm
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bq
29$D_{4}$ \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_w
31$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.31.am_da
41$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_g
43$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_cs
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_dy
59$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_ag
61$D_{4}$ \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.61.m_da
67$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_cs
71$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_da
73$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_fm
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.79.a_da
83$D_{4}$ \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.83.au_jm
89$D_{4}$ \( 1 + 22 T + 254 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.89.w_ju
97$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.97.a_o
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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

−8.782028571213032384292033192455, −8.275363247006645084219880716051, −8.162796578461588944914250778936, −8.044976487369645510405441228168, −7.61181443390912998048363527990, −7.59621024735117341019591045294, −6.86222591641897276492704544030, −6.28721564626949737831762944589, −5.90243686021373011354021344874, −5.39244912607960534630250353185, −4.92010125047906564554295498789, −4.85566696244372164831932494478, −4.34331218097168279261561274145, −4.15631154445076857531036946173, −3.18662125245793561674839628210, −3.04673032684890458268398583507, −2.26793858360261795175298145794, −1.79305340950460555158943807143, −1.24958232241972367064256834647, −0.65537376561778620533501842580, 0.65537376561778620533501842580, 1.24958232241972367064256834647, 1.79305340950460555158943807143, 2.26793858360261795175298145794, 3.04673032684890458268398583507, 3.18662125245793561674839628210, 4.15631154445076857531036946173, 4.34331218097168279261561274145, 4.85566696244372164831932494478, 4.92010125047906564554295498789, 5.39244912607960534630250353185, 5.90243686021373011354021344874, 6.28721564626949737831762944589, 6.86222591641897276492704544030, 7.59621024735117341019591045294, 7.61181443390912998048363527990, 8.044976487369645510405441228168, 8.162796578461588944914250778936, 8.275363247006645084219880716051, 8.782028571213032384292033192455

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