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Properties

Label 4-59456-1.1-c1e2-0-2
Degree $4$
Conductor $59456$
Sign $-1$
Analytic cond. $3.79096$
Root an. cond. $1.39536$
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 + 3·3-s − 4-s − 3·6-s + 3·8-s + 9-s − 8·11-s − 3·12-s − 16-s − 3·17-s − 18-s − 2·19-s + 8·22-s + 9·24-s − 3·25-s − 12·27-s − 5·32-s − 24·33-s + 3·34-s − 36-s + 2·38-s + 41-s + 5·43-s + 8·44-s − 3·48-s + 2·49-s + 3·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s − 1/2·4-s − 1.22·6-s + 1.06·8-s + 1/3·9-s − 2.41·11-s − 0.866·12-s − 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.458·19-s + 1.70·22-s + 1.83·24-s − 3/5·25-s − 2.30·27-s − 0.883·32-s − 4.17·33-s + 0.514·34-s − 1/6·36-s + 0.324·38-s + 0.156·41-s + 0.762·43-s + 1.20·44-s − 0.433·48-s + 2/7·49-s + 0.424·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(59456\)    =    \(2^{6} \cdot 929\)
Sign: $-1$
Analytic conductor: \(3.79096\)
Root analytic conductor: \(1.39536\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 59456,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
929$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 12 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.3.ad_i
5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.5.a_d
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.i_bi
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.13.a_i
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.19.c_bj
23$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.23.a_abh
29$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \) 2.29.a_bl
31$C_2^2$ \( 1 + 48 T^{2} + p^{2} T^{4} \) 2.31.a_bw
37$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.37.a_cd
41$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.ab_bo
43$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.af_di
47$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.47.a_l
53$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.53.a_k
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.d_k
61$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \) 2.61.a_bv
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ak_da
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.71.a_aec
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.73.n_ha
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.79.a_fa
83$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.83.e_ba
89$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.89.at_jw
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.i_hy
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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

−9.351179470708541792817771152731, −9.262810853862041235280162353330, −8.598289220694523607984088980359, −8.192689607377138028831345231778, −7.969809754561556987472941512407, −7.57210176814140809418140390839, −6.91415544193008153035948322958, −5.81333205091838344570577985801, −5.44988878342041970039981388786, −4.67616867410210482142923725030, −4.00631027382538882523488911907, −3.19490811001888482832396836601, −2.54659819748699058583971187462, −2.08028249559320308149877439936, 0, 2.08028249559320308149877439936, 2.54659819748699058583971187462, 3.19490811001888482832396836601, 4.00631027382538882523488911907, 4.67616867410210482142923725030, 5.44988878342041970039981388786, 5.81333205091838344570577985801, 6.91415544193008153035948322958, 7.57210176814140809418140390839, 7.969809754561556987472941512407, 8.192689607377138028831345231778, 8.598289220694523607984088980359, 9.262810853862041235280162353330, 9.351179470708541792817771152731

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