Properties

Label 4-10e4-1.1-c1e2-0-1
Degree $4$
Conductor $10000$
Sign $1$
Analytic cond. $0.637608$
Root an. cond. $0.893590$
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  + 4·11-s + 4·31-s − 16·41-s − 16·61-s + 4·71-s − 9·81-s + 101-s + 103-s + 107-s + 109-s + 113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 1.20·11-s + 0.718·31-s − 2.49·41-s − 2.04·61-s + 0.474·71-s − 81-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.909·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10000\)    =    \(2^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.637608\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.031407104\)
\(L(\frac12)\) \(\approx\) \(1.031407104\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
7$C_2^2$ \( 1 + p^{2} T^{4} \)
11$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + p^{2} T^{4} \)
17$C_2^2$ \( 1 + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_4$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + p^{2} T^{4} \)
41$C_4$ \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + p^{2} T^{4} \)
47$C_2^2$ \( 1 + p^{2} T^{4} \)
53$C_2^2$ \( 1 + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_4$ \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + p^{2} T^{4} \)
71$C_4$ \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + p^{2} T^{4} \)
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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

−16.7291169704, −16.1445597827, −15.4652195600, −15.1981842665, −14.6390108237, −14.0157809410, −13.7591239186, −13.1427884619, −12.4954271715, −11.9742485394, −11.6287241407, −11.0536414542, −10.2888508405, −9.95018153115, −9.16859678328, −8.79007106897, −8.12054788687, −7.44064843243, −6.65487342490, −6.33801339792, −5.40492896827, −4.64669821940, −3.86402810062, −3.01417506275, −1.62086060708, 1.62086060708, 3.01417506275, 3.86402810062, 4.64669821940, 5.40492896827, 6.33801339792, 6.65487342490, 7.44064843243, 8.12054788687, 8.79007106897, 9.16859678328, 9.95018153115, 10.2888508405, 11.0536414542, 11.6287241407, 11.9742485394, 12.4954271715, 13.1427884619, 13.7591239186, 14.0157809410, 14.6390108237, 15.1981842665, 15.4652195600, 16.1445597827, 16.7291169704

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