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Properties

Label 4-2664e2-1.1-c1e2-0-8
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 7-s + 4·11-s − 3·13-s + 8·17-s + 8·19-s + 5·25-s + 20·29-s − 14·31-s − 4·35-s + 10·37-s − 2·41-s + 10·43-s + 12·47-s + 7·49-s + 6·53-s + 16·55-s − 12·59-s + 2·61-s − 12·65-s − 9·67-s + 10·71-s − 22·73-s − 4·77-s + 9·79-s − 12·83-s + 32·85-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.377·7-s + 1.20·11-s − 0.832·13-s + 1.94·17-s + 1.83·19-s + 25-s + 3.71·29-s − 2.51·31-s − 0.676·35-s + 1.64·37-s − 0.312·41-s + 1.52·43-s + 1.75·47-s + 49-s + 0.824·53-s + 2.15·55-s − 1.56·59-s + 0.256·61-s − 1.48·65-s − 1.09·67-s + 1.18·71-s − 2.57·73-s − 0.455·77-s + 1.01·79-s − 1.31·83-s + 3.47·85-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

\(L(1)\) \(\approx\) \(5.464261798\)
\(L(\frac12)\) \(\approx\) \(5.464261798\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 10 T + p T^{2} \)
good5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_l
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.b_ag
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.13.d_ae
17$C_2^2$ \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bv
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.19.ai_bt
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.29.au_gc
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.31.o_eh
41$C_2^2$ \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.41.c_abl
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.43.ak_eh
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_acf
67$C_2^2$ \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.67.j_o
71$C_2^2$ \( 1 - 10 T + 29 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.71.ak_bd
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.73.w_kh
79$C_2^2$ \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.79.aj_c
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_acb
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.97.ac_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

−9.255865215419228105865534724722, −8.930267028895457925206521863206, −8.240222300797045691805835038423, −7.921425022210030726747763241306, −7.28531753485708225392702017273, −7.24463525534480450554842888535, −6.79493446568862523005301218183, −6.21488307505504371887496639473, −5.77703665255426160885376950029, −5.71890343084899947023491434100, −5.40235722226429424118233048168, −4.80472964170495918478561105205, −4.25066591856404981381495416740, −3.93087850472425164840570110962, −3.09512398078593486238080053431, −2.91872214889269100483356165941, −2.49516652736310633098288865380, −1.75441238870526024657891730155, −1.10536310643562621886973128308, −0.958811296649683288568431417256, 0.958811296649683288568431417256, 1.10536310643562621886973128308, 1.75441238870526024657891730155, 2.49516652736310633098288865380, 2.91872214889269100483356165941, 3.09512398078593486238080053431, 3.93087850472425164840570110962, 4.25066591856404981381495416740, 4.80472964170495918478561105205, 5.40235722226429424118233048168, 5.71890343084899947023491434100, 5.77703665255426160885376950029, 6.21488307505504371887496639473, 6.79493446568862523005301218183, 7.24463525534480450554842888535, 7.28531753485708225392702017273, 7.921425022210030726747763241306, 8.240222300797045691805835038423, 8.930267028895457925206521863206, 9.255865215419228105865534724722

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