Properties

Label 4-275e2-1.1-c1e2-0-7
Degree $4$
Conductor $75625$
Sign $1$
Analytic cond. $4.82191$
Root an. cond. $1.48185$
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  + 3·4-s + 6·9-s − 2·11-s + 5·16-s + 8·19-s − 12·29-s − 16·31-s + 18·36-s + 4·41-s − 6·44-s + 14·49-s − 8·59-s − 20·61-s + 3·64-s + 16·71-s + 24·76-s − 16·79-s + 27·81-s − 20·89-s − 12·99-s − 20·101-s + 36·109-s − 36·116-s + 3·121-s − 48·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 3/2·4-s + 2·9-s − 0.603·11-s + 5/4·16-s + 1.83·19-s − 2.22·29-s − 2.87·31-s + 3·36-s + 0.624·41-s − 0.904·44-s + 2·49-s − 1.04·59-s − 2.56·61-s + 3/8·64-s + 1.89·71-s + 2.75·76-s − 1.80·79-s + 3·81-s − 2.11·89-s − 1.20·99-s − 1.99·101-s + 3.44·109-s − 3.34·116-s + 3/11·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(75625\)    =    \(5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4.82191\)
Root analytic conductor: \(1.48185\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 75625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.370225828\)
\(L(\frac12)\) \(\approx\) \(2.370225828\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−12.20675355460443389363383181244, −11.56935558661056245702572763122, −11.00865143116840149642500161602, −10.90794687354454144642014721749, −10.35651288740213495530735293664, −9.792268267361740489889061113069, −9.350269247137370398911964447819, −9.037887197330262027216882065514, −7.86650490421446811802287575091, −7.48769381985499203074271236958, −7.24996889059642868313010814463, −7.08794643128684311937364276526, −6.14066830454167328996642206422, −5.59457055083573837498819553075, −5.19694025163837720473738220247, −4.19185147450923176219971115246, −3.66850955045543300252274607757, −2.93007012260954588702810304303, −1.93823042745280027039669861818, −1.48799635321176176224017572114, 1.48799635321176176224017572114, 1.93823042745280027039669861818, 2.93007012260954588702810304303, 3.66850955045543300252274607757, 4.19185147450923176219971115246, 5.19694025163837720473738220247, 5.59457055083573837498819553075, 6.14066830454167328996642206422, 7.08794643128684311937364276526, 7.24996889059642868313010814463, 7.48769381985499203074271236958, 7.86650490421446811802287575091, 9.037887197330262027216882065514, 9.350269247137370398911964447819, 9.792268267361740489889061113069, 10.35651288740213495530735293664, 10.90794687354454144642014721749, 11.00865143116840149642500161602, 11.56935558661056245702572763122, 12.20675355460443389363383181244

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