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Properties

Label 4-264e2-1.1-c1e2-0-39
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 4-s + 6-s − 3·8-s − 2·9-s − 2·11-s − 12-s − 16-s − 7·17-s − 2·18-s − 3·19-s − 2·22-s − 3·24-s − 2·25-s − 5·27-s + 5·32-s − 2·33-s − 7·34-s + 2·36-s − 3·38-s + 9·41-s − 13·43-s + 2·44-s − 48-s − 12·49-s − 2·50-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s − 2/3·9-s − 0.603·11-s − 0.288·12-s − 1/4·16-s − 1.69·17-s − 0.471·18-s − 0.688·19-s − 0.426·22-s − 0.612·24-s − 2/5·25-s − 0.962·27-s + 0.883·32-s − 0.348·33-s − 1.20·34-s + 1/3·36-s − 0.486·38-s + 1.40·41-s − 1.98·43-s + 0.301·44-s − 0.144·48-s − 1.71·49-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: \(1\)
Selberg data: \((4,\ 69696,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
3$C_2$ \( 1 - T + p T^{2} \)
11$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.7.a_m
13$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.13.a_t
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.h_bi
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.d_bc
23$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.23.a_n
29$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.29.a_bg
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.37.a_abi
41$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.aj_de
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.n_du
47$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.47.a_abu
53$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.53.a_ca
59$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.59.ai_fd
61$C_2^2$ \( 1 - 75 T^{2} + p^{2} T^{4} \) 2.61.a_acx
67$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.f_eg
71$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.71.a_dm
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.73.ai_fq
79$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.79.a_adc
83$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.83.au_iw
89$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.89.l_hy
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.97.ad_ho
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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

−9.608819087779611212222544944177, −8.911016915139651177261520974791, −8.686693585152316872360977257552, −8.141806743752138935353208554587, −7.75975577794045985195144717441, −6.84729454041940662816327647768, −6.36199530692318522192293276709, −5.89949320206501939586597679095, −5.10632291865799752593737816190, −4.77111876658569080820894696581, −4.02827327581752341264840142476, −3.47502779919987581669338220749, −2.69772410444506472373161239464, −2.09433173010221596933601076381, 0, 2.09433173010221596933601076381, 2.69772410444506472373161239464, 3.47502779919987581669338220749, 4.02827327581752341264840142476, 4.77111876658569080820894696581, 5.10632291865799752593737816190, 5.89949320206501939586597679095, 6.36199530692318522192293276709, 6.84729454041940662816327647768, 7.75975577794045985195144717441, 8.141806743752138935353208554587, 8.686693585152316872360977257552, 8.911016915139651177261520974791, 9.608819087779611212222544944177

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