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Properties

Label 4-264e2-1.1-c1e2-0-4
Degree $4$
Conductor $69696$
Sign $1$
Analytic cond. $4.44387$
Root an. cond. $1.45191$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 4-s − 2·6-s − 3·8-s + 9-s − 4·11-s + 2·12-s − 16-s + 2·17-s + 18-s + 2·19-s − 4·22-s + 6·24-s − 2·25-s + 4·27-s + 5·32-s + 8·33-s + 2·34-s − 36-s + 2·38-s − 4·41-s + 8·43-s + 4·44-s + 2·48-s + 10·49-s − 2·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.816·6-s − 1.06·8-s + 1/3·9-s − 1.20·11-s + 0.577·12-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.458·19-s − 0.852·22-s + 1.22·24-s − 2/5·25-s + 0.769·27-s + 0.883·32-s + 1.39·33-s + 0.342·34-s − 1/6·36-s + 0.324·38-s − 0.624·41-s + 1.21·43-s + 0.603·44-s + 0.288·48-s + 10/7·49-s − 0.282·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(69696\)    =    \(2^{6} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4.44387\)
Root analytic conductor: \(1.45191\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 69696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8257568917\)
\(L(\frac12)\) \(\approx\) \(0.8257568917\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.13.a_i
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.ac_k
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.19.ac_bm
23$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.23.a_au
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.29.a_k
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.e_cs
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.47.a_acq
53$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.53.a_acg
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ac_dq
61$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.61.a_cq
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.71.a_cq
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.am_eo
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.79.a_ao
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.e_gk
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.89.aw_le
97$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) 2.97.abc_ow
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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

−10.02620030027267293647816707089, −9.355894338711399096917542908235, −8.902083907104384842398100564511, −8.312738125827567261208082659778, −7.70109251877832870529048998824, −7.28261519323983315635741168656, −6.45570148983262156095847316778, −5.91921270867214628239046225141, −5.60486434057152358801045736548, −5.01975876943164840903559739842, −4.69790409327146333552953896476, −3.84018274550088006229899004933, −3.17878844714098546612852079072, −2.35109818535381695130055222176, −0.68608367511705435793727716287, 0.68608367511705435793727716287, 2.35109818535381695130055222176, 3.17878844714098546612852079072, 3.84018274550088006229899004933, 4.69790409327146333552953896476, 5.01975876943164840903559739842, 5.60486434057152358801045736548, 5.91921270867214628239046225141, 6.45570148983262156095847316778, 7.28261519323983315635741168656, 7.70109251877832870529048998824, 8.312738125827567261208082659778, 8.902083907104384842398100564511, 9.355894338711399096917542908235, 10.02620030027267293647816707089

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