Properties

Label 4-304e2-1.1-c1e2-0-10
Degree $4$
Conductor $92416$
Sign $1$
Analytic cond. $5.89252$
Root an. cond. $1.55802$
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  + 4·5-s + 3·9-s + 2·17-s + 2·25-s + 12·45-s − 5·49-s + 22·73-s + 8·85-s − 20·101-s − 18·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 6·153-s + 157-s + 163-s + 167-s + 17·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 1.78·5-s + 9-s + 0.485·17-s + 2/5·25-s + 1.78·45-s − 5/7·49-s + 2.57·73-s + 0.867·85-s − 1.99·101-s − 1.63·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.485·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(92416\)    =    \(2^{8} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5.89252\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 92416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.276899970\)
\(L(\frac12)\) \(\approx\) \(2.276899970\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
19$C_2$ \( 1 + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−9.711126156416052443098396228470, −9.315762093869155493529849873103, −8.848691784087101181992655860700, −7.969052305916730780775791302254, −7.78719480788708539513415919699, −6.92890982290269651491366057507, −6.54133093505676792175277300295, −6.07986232560123820629791246302, −5.40224276193534648970565397003, −5.14918115004471567068095677227, −4.26420334333379584262856788454, −3.66768334138438131592754563716, −2.69966505038669909473735177757, −1.98449756726674810594162410661, −1.33788236914624712075940377525, 1.33788236914624712075940377525, 1.98449756726674810594162410661, 2.69966505038669909473735177757, 3.66768334138438131592754563716, 4.26420334333379584262856788454, 5.14918115004471567068095677227, 5.40224276193534648970565397003, 6.07986232560123820629791246302, 6.54133093505676792175277300295, 6.92890982290269651491366057507, 7.78719480788708539513415919699, 7.969052305916730780775791302254, 8.848691784087101181992655860700, 9.315762093869155493529849873103, 9.711126156416052443098396228470

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