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Properties

Label 4-92e4-1.1-c1e2-0-13
Degree $4$
Conductor $71639296$
Sign $1$
Analytic cond. $4567.78$
Root an. cond. $8.22103$
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 + 4·7-s − 4·9-s − 6·13-s + 4·19-s + 25-s + 6·29-s − 12·31-s − 8·35-s + 8·37-s + 14·41-s + 12·43-s + 8·45-s − 14·53-s − 4·59-s − 2·61-s − 16·63-s + 12·65-s − 8·67-s + 4·71-s − 2·73-s − 8·79-s + 7·81-s − 20·83-s − 6·89-s − 24·91-s − 8·95-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s − 4/3·9-s − 1.66·13-s + 0.917·19-s + 1/5·25-s + 1.11·29-s − 2.15·31-s − 1.35·35-s + 1.31·37-s + 2.18·41-s + 1.82·43-s + 1.19·45-s − 1.92·53-s − 0.520·59-s − 0.256·61-s − 2.01·63-s + 1.48·65-s − 0.977·67-s + 0.474·71-s − 0.234·73-s − 0.900·79-s + 7/9·81-s − 2.19·83-s − 0.635·89-s − 2.51·91-s − 0.820·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71639296\)    =    \(2^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(4567.78\)
Root analytic conductor: \(8.22103\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 71639296,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
23 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_d
7$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_q
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.11.a_u
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.13.g_bj
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.17.a_ba
19$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_y
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.31.m_du
37$D_{4}$ \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_cg
41$D_{4}$ \( 1 - 14 T + 3 p T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.41.ao_et
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.43.am_es
47$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.47.a_bs
53$D_{4}$ \( 1 + 14 T + 147 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.53.o_fr
59$D_{4}$ \( 1 + 4 T + 120 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_eq
61$D_{4}$ \( 1 + 2 T + 91 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_dn
67$D_{4}$ \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_w
71$D_{4}$ \( 1 - 4 T + 96 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_ds
73$D_{4}$ \( 1 + 2 T + 139 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_fj
79$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_fm
83$D_{4}$ \( 1 + 20 T + 234 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.83.u_ja
89$D_{4}$ \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_fz
97$D_{4}$ \( 1 - 10 T + 147 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_fr
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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

−7.54200061416706413533794063835, −7.50874829121518353165241884472, −7.22244988842786564487293060540, −6.61352540912707418444380443015, −6.10358035863667193735429526056, −5.80659716340046450683004002955, −5.57540053956246072112418951471, −5.14332871247750322455392712715, −4.63444850735236045105454576394, −4.56773644710538837415838381282, −4.32590002931990910461433092958, −3.52892290302461112451865494721, −3.38163071282561998056299374169, −2.74368143606713615755657805109, −2.43974391005491642025762445668, −2.23139290360532520884205941446, −1.24898941727549985984316081126, −1.17426576279374763593641468841, 0, 0, 1.17426576279374763593641468841, 1.24898941727549985984316081126, 2.23139290360532520884205941446, 2.43974391005491642025762445668, 2.74368143606713615755657805109, 3.38163071282561998056299374169, 3.52892290302461112451865494721, 4.32590002931990910461433092958, 4.56773644710538837415838381282, 4.63444850735236045105454576394, 5.14332871247750322455392712715, 5.57540053956246072112418951471, 5.80659716340046450683004002955, 6.10358035863667193735429526056, 6.61352540912707418444380443015, 7.22244988842786564487293060540, 7.50874829121518353165241884472, 7.54200061416706413533794063835

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