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Properties

Label 4-96e2-1.1-c1e2-0-5
Degree $4$
Conductor $9216$
Sign $1$
Analytic cond. $0.587620$
Root an. cond. $0.875536$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 9-s − 4·13-s − 12·17-s + 2·25-s + 4·29-s − 4·37-s + 4·41-s + 4·45-s + 2·49-s + 20·53-s + 12·61-s − 16·65-s − 12·73-s + 81-s − 48·85-s + 20·89-s − 28·97-s − 12·101-s + 28·109-s + 4·113-s − 4·117-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.78·5-s + 1/3·9-s − 1.10·13-s − 2.91·17-s + 2/5·25-s + 0.742·29-s − 0.657·37-s + 0.624·41-s + 0.596·45-s + 2/7·49-s + 2.74·53-s + 1.53·61-s − 1.98·65-s − 1.40·73-s + 1/9·81-s − 5.20·85-s + 2.11·89-s − 2.84·97-s − 1.19·101-s + 2.68·109-s + 0.376·113-s − 0.369·117-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9216\)    =    \(2^{10} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(0.587620\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9216,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.172682149\)
\(L(\frac12)\) \(\approx\) \(1.172682149\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.53.au_hy
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.a_aek
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−11.67525385332011727244436354557, −10.84849442743057013529606410604, −10.44913304188171401892835914004, −9.826637660192491869357018654726, −9.488699237661810053920246470193, −8.839886292016704738441339481672, −8.391046533691483698004382219288, −7.17212008872630953090710242182, −6.91469103626909406223255043804, −6.15267013015603292112906972931, −5.56983356275681900785379754785, −4.77602747706575131165960838209, −4.10625798014363516984369255368, −2.44821729992400328434125228202, −2.09803110754033007053716721218, 2.09803110754033007053716721218, 2.44821729992400328434125228202, 4.10625798014363516984369255368, 4.77602747706575131165960838209, 5.56983356275681900785379754785, 6.15267013015603292112906972931, 6.91469103626909406223255043804, 7.17212008872630953090710242182, 8.391046533691483698004382219288, 8.839886292016704738441339481672, 9.488699237661810053920246470193, 9.826637660192491869357018654726, 10.44913304188171401892835914004, 10.84849442743057013529606410604, 11.67525385332011727244436354557

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