Properties

Label 4-304e2-1.1-c1e2-0-3
Degree $4$
Conductor $92416$
Sign $1$
Analytic cond. $5.89252$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 6·9-s + 14·17-s − 7·25-s + 12·45-s − 5·49-s + 30·61-s + 22·73-s + 27·81-s − 28·85-s − 20·101-s + 3·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 84·153-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 0.894·5-s − 2·9-s + 3.39·17-s − 7/5·25-s + 1.78·45-s − 5/7·49-s + 3.84·61-s + 2.57·73-s + 3·81-s − 3.03·85-s − 1.99·101-s + 3/11·121-s + 2.32·125-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 − 6.79·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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: induced by $\chi_{304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 92416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.007175762\)
\(L(\frac12)\) \(\approx\) \(1.007175762\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−11.93844245095129171959488032957, −11.37058711974801084999360835271, −11.36424100874035080602044629917, −10.64039820529012766146601931497, −9.865224886415717308099031813263, −9.810156314211485736748144542945, −9.169588155260978087686422243889, −8.246007557642040916600789169367, −8.176959562550778912654720271459, −7.935279911670733220867378369606, −7.25110734562622309623475704069, −6.57397978870308106344754277290, −5.76173891115666305871457994825, −5.54429051267460212401686320132, −5.17400013551212808950091505439, −4.04036788852722839709519649446, −3.45263935766864084060798214387, −3.18985105245041001800734306256, −2.20554819837438602305096567011, −0.75067994626145005201323780568, 0.75067994626145005201323780568, 2.20554819837438602305096567011, 3.18985105245041001800734306256, 3.45263935766864084060798214387, 4.04036788852722839709519649446, 5.17400013551212808950091505439, 5.54429051267460212401686320132, 5.76173891115666305871457994825, 6.57397978870308106344754277290, 7.25110734562622309623475704069, 7.935279911670733220867378369606, 8.176959562550778912654720271459, 8.246007557642040916600789169367, 9.169588155260978087686422243889, 9.810156314211485736748144542945, 9.865224886415717308099031813263, 10.64039820529012766146601931497, 11.36424100874035080602044629917, 11.37058711974801084999360835271, 11.93844245095129171959488032957

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