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Properties

Label 4-975e2-1.1-c0e2-0-5
Degree $4$
Conductor $950625$
Sign $1$
Analytic cond. $0.236768$
Root an. cond. $0.697558$
Motivic weight $0$
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  + 2·3-s + 3·9-s − 2·13-s − 16-s + 4·27-s − 4·39-s − 2·48-s + 2·49-s + 5·81-s − 6·117-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2·3-s + 3·9-s − 2·13-s − 16-s + 4·27-s − 4·39-s − 2·48-s + 2·49-s + 5·81-s − 6·117-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(950625\)    =    \(3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.236768\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 950625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.848377800\)
\(L(\frac12)\) \(\approx\) \(1.848377800\)
\(L(1)\) 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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2^2$ \( 1 + T^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−10.17934329951944876245294953206, −9.973438318531000944450321310781, −9.445482340411059171270096194668, −9.083492567683835646843068218470, −8.915114406647258837726466396188, −8.407566945802334014804710733946, −7.82001538497289228716159363266, −7.58543290177525607492175639340, −7.24588872077008628092812186312, −6.80616561924932108697696819786, −6.44021501339372778440570961299, −5.57818974090550141849119573935, −4.84824727925279207794426501916, −4.77124347468273750850239592743, −3.98677422463913272559462774347, −3.77565726747821271093854660763, −2.90525958001038370072008469250, −2.43833327113116172909390124467, −2.30666693957681143269879376228, −1.37193100007792083659560202861, 1.37193100007792083659560202861, 2.30666693957681143269879376228, 2.43833327113116172909390124467, 2.90525958001038370072008469250, 3.77565726747821271093854660763, 3.98677422463913272559462774347, 4.77124347468273750850239592743, 4.84824727925279207794426501916, 5.57818974090550141849119573935, 6.44021501339372778440570961299, 6.80616561924932108697696819786, 7.24588872077008628092812186312, 7.58543290177525607492175639340, 7.82001538497289228716159363266, 8.407566945802334014804710733946, 8.915114406647258837726466396188, 9.083492567683835646843068218470, 9.445482340411059171270096194668, 9.973438318531000944450321310781, 10.17934329951944876245294953206

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