Properties

Label 4-1950e2-1.1-c1e2-0-9
Degree $4$
Conductor $3802500$
Sign $1$
Analytic cond. $242.450$
Root an. cond. $3.94598$
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  − 2-s + 3-s − 6-s − 5·7-s + 8-s + 3·11-s + 5·13-s + 5·14-s − 16-s − 8·17-s + 5·19-s − 5·21-s − 3·22-s − 4·23-s + 24-s − 5·26-s − 27-s + 4·29-s − 4·31-s + 3·33-s + 8·34-s − 7·37-s − 5·38-s + 5·39-s − 6·41-s + 5·42-s + 6·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 0.408·6-s − 1.88·7-s + 0.353·8-s + 0.904·11-s + 1.38·13-s + 1.33·14-s − 1/4·16-s − 1.94·17-s + 1.14·19-s − 1.09·21-s − 0.639·22-s − 0.834·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s + 0.742·29-s − 0.718·31-s + 0.522·33-s + 1.37·34-s − 1.15·37-s − 0.811·38-s + 0.800·39-s − 0.937·41-s + 0.771·42-s + 0.914·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3802500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(242.450\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1950} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3802500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7775115140\)
\(L(\frac12)\) \(\approx\) \(0.7775115140\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 - 5 T + p T^{2} \)
good7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 2 T - 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 11 T + 32 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
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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

−9.282476357624505982902813602347, −8.872802615036259841930815296848, −8.782708619609935481433878211102, −8.556830018957462287171729480271, −7.83540007868342819771502571880, −7.41971767222240864084276215627, −7.11331747970344182469529886768, −6.50574419344820441035150218202, −6.35288119794000054171307021912, −6.11495133337382610329286559508, −5.49887297664766994310760049203, −4.84756799930984082284646857199, −4.34516099912894322106774768519, −3.71767503134361079224763046582, −3.63152583939253463928614654608, −3.10113767207549748088282599526, −2.54627793161677866062291266947, −1.86280931083554841523579625705, −1.28770776188150996851293617787, −0.36417996033996846818992111353, 0.36417996033996846818992111353, 1.28770776188150996851293617787, 1.86280931083554841523579625705, 2.54627793161677866062291266947, 3.10113767207549748088282599526, 3.63152583939253463928614654608, 3.71767503134361079224763046582, 4.34516099912894322106774768519, 4.84756799930984082284646857199, 5.49887297664766994310760049203, 6.11495133337382610329286559508, 6.35288119794000054171307021912, 6.50574419344820441035150218202, 7.11331747970344182469529886768, 7.41971767222240864084276215627, 7.83540007868342819771502571880, 8.556830018957462287171729480271, 8.782708619609935481433878211102, 8.872802615036259841930815296848, 9.282476357624505982902813602347

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