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Properties

Label 4-7616-1.1-c1e2-0-3
Degree $4$
Conductor $7616$
Sign $-1$
Analytic cond. $0.485602$
Root an. cond. $0.834776$
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  − 2·2-s + 2·4-s − 5·7-s − 4·9-s + 10·14-s − 4·16-s + 17-s + 8·18-s − 10·23-s − 4·25-s − 10·28-s − 2·31-s + 8·32-s − 2·34-s − 8·36-s + 4·41-s + 20·46-s + 6·47-s + 14·49-s + 8·50-s + 4·62-s + 20·63-s − 8·64-s + 2·68-s − 24·71-s + 6·73-s − 4·79-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.88·7-s − 4/3·9-s + 2.67·14-s − 16-s + 0.242·17-s + 1.88·18-s − 2.08·23-s − 4/5·25-s − 1.88·28-s − 0.359·31-s + 1.41·32-s − 0.342·34-s − 4/3·36-s + 0.624·41-s + 2.94·46-s + 0.875·47-s + 2·49-s + 1.13·50-s + 0.508·62-s + 2.51·63-s − 64-s + 0.242·68-s − 2.84·71-s + 0.702·73-s − 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7616\)    =    \(2^{6} \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(0.485602\)
Root analytic conductor: \(0.834776\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 7616,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 4 T + p T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.k_cs
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.29.a_abm
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.c_as
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.37.a_c
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ae_cs
43$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.43.a_g
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.ag_da
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.53.a_cg
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.59.a_ak
61$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.61.a_abg
67$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.67.a_acg
71$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.y_kk
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.ag_fa
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.e_ew
83$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.83.a_abi
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.m_hq
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.97.ay_mw
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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

−11.55032917911083036223273774314, −10.65957203790405176344714507266, −10.13029789551604852745761945506, −9.869554446902281522821813817765, −9.059545299503891983579368515498, −8.921729762850024952847345387493, −8.020474463866494776082197407304, −7.58950700040431763777317518772, −6.76994087958107150520019013019, −6.01827375919118557947246085328, −5.74788901826129237856026285973, −4.22292289409310699328837314231, −3.30108762319793815148919643227, −2.32246137883598016340465501420, 0, 2.32246137883598016340465501420, 3.30108762319793815148919643227, 4.22292289409310699328837314231, 5.74788901826129237856026285973, 6.01827375919118557947246085328, 6.76994087958107150520019013019, 7.58950700040431763777317518772, 8.020474463866494776082197407304, 8.921729762850024952847345387493, 9.059545299503891983579368515498, 9.869554446902281522821813817765, 10.13029789551604852745761945506, 10.65957203790405176344714507266, 11.55032917911083036223273774314

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