Properties

Label 4-1900e2-1.1-c1e2-0-1
Degree $4$
Conductor $3610000$
Sign $1$
Analytic cond. $230.176$
Root an. cond. $3.89507$
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  + 2·9-s + 2·19-s + 4·29-s + 8·31-s − 20·41-s + 10·49-s + 8·59-s + 4·61-s + 24·71-s − 16·79-s − 5·81-s − 4·89-s − 20·101-s − 20·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 4·171-s + 173-s + ⋯
L(s)  = 1  + 2/3·9-s + 0.458·19-s + 0.742·29-s + 1.43·31-s − 3.12·41-s + 10/7·49-s + 1.04·59-s + 0.512·61-s + 2.84·71-s − 1.80·79-s − 5/9·81-s − 0.423·89-s − 1.99·101-s − 1.91·109-s − 2·121-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 − 0.769·169-s + 0.305·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3610000\)    =    \(2^{4} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(230.176\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3610000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.425877383\)
\(L(\frac12)\) \(\approx\) \(2.425877383\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−9.589794720351627286948179479459, −8.902395607164004577111770790657, −8.604157004218894647066953018032, −8.215003478037161535612588454734, −8.006229235820168198037045670736, −7.38085164086044896113199831655, −6.95102633656829355135763511278, −6.68623316230868009385269762015, −6.45540485107953424439008618878, −5.73408840322144802552398059789, −5.25878705003116847708138839491, −5.08425147123881701964406324837, −4.49975178305613840369240429382, −3.88833863067598626535280703104, −3.80027802744000710567578750861, −2.83337787832829236963395980921, −2.75708110867717122872784250534, −1.85608172423703575011480547082, −1.35144508543905545735368819591, −0.60562547916436570633318595538, 0.60562547916436570633318595538, 1.35144508543905545735368819591, 1.85608172423703575011480547082, 2.75708110867717122872784250534, 2.83337787832829236963395980921, 3.80027802744000710567578750861, 3.88833863067598626535280703104, 4.49975178305613840369240429382, 5.08425147123881701964406324837, 5.25878705003116847708138839491, 5.73408840322144802552398059789, 6.45540485107953424439008618878, 6.68623316230868009385269762015, 6.95102633656829355135763511278, 7.38085164086044896113199831655, 8.006229235820168198037045670736, 8.215003478037161535612588454734, 8.604157004218894647066953018032, 8.902395607164004577111770790657, 9.589794720351627286948179479459

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