Properties

Label 4-140e2-1.1-c1e2-0-1
Degree $4$
Conductor $19600$
Sign $1$
Analytic cond. $1.24971$
Root an. cond. $1.05731$
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  − 4·5-s − 3·9-s + 6·11-s + 16·19-s + 11·25-s + 2·29-s − 4·31-s − 12·41-s + 12·45-s − 49-s − 24·55-s + 20·59-s + 14·79-s − 16·89-s − 64·95-s − 18·99-s − 24·101-s + 14·109-s + 5·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.78·5-s − 9-s + 1.80·11-s + 3.67·19-s + 11/5·25-s + 0.371·29-s − 0.718·31-s − 1.87·41-s + 1.78·45-s − 1/7·49-s − 3.23·55-s + 2.60·59-s + 1.57·79-s − 1.69·89-s − 6.56·95-s − 1.80·99-s − 2.38·101-s + 1.34·109-s + 5/11·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.24971\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 19600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8976186430\)
\(L(\frac12)\) \(\approx\) \(0.8976186430\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 27 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 185 T^{2} + p^{2} T^{4} \)
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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

−13.88118211323064041135408829373, −12.71563600391948896263354581328, −12.06105169892242275856953944750, −11.76881735356402041011823007525, −11.58313917389268633122865530347, −11.27725875409626222898327406774, −10.43711882064880040036960835280, −9.516537308298576046776730123468, −9.420974907329973426166595707872, −8.493809455508208449680392700656, −8.314702116307900173717196464658, −7.47238737875400438444149209713, −7.10639162833427225678781300528, −6.58387417840574902412583242214, −5.48358601022321794975607084254, −5.13603431928211944546808052890, −4.07595851152587374114167581579, −3.50118312923384028984960173143, −3.07637493591239790399473722958, −1.09919920217323917967192336340, 1.09919920217323917967192336340, 3.07637493591239790399473722958, 3.50118312923384028984960173143, 4.07595851152587374114167581579, 5.13603431928211944546808052890, 5.48358601022321794975607084254, 6.58387417840574902412583242214, 7.10639162833427225678781300528, 7.47238737875400438444149209713, 8.314702116307900173717196464658, 8.493809455508208449680392700656, 9.420974907329973426166595707872, 9.516537308298576046776730123468, 10.43711882064880040036960835280, 11.27725875409626222898327406774, 11.58313917389268633122865530347, 11.76881735356402041011823007525, 12.06105169892242275856953944750, 12.71563600391948896263354581328, 13.88118211323064041135408829373

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