Properties

Label 4-1932e2-1.1-c1e2-0-8
Degree $4$
Conductor $3732624$
Sign $1$
Analytic cond. $237.995$
Root an. cond. $3.92773$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5·5-s + 2·7-s + 3·9-s − 2·11-s − 13-s − 10·15-s − 10·17-s + 2·19-s + 4·21-s + 2·23-s + 10·25-s + 4·27-s − 8·29-s − 4·31-s − 4·33-s − 10·35-s − 6·37-s − 2·39-s − 16·41-s − 3·43-s − 15·45-s − 6·47-s + 3·49-s − 20·51-s + 3·53-s + 10·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 2.23·5-s + 0.755·7-s + 9-s − 0.603·11-s − 0.277·13-s − 2.58·15-s − 2.42·17-s + 0.458·19-s + 0.872·21-s + 0.417·23-s + 2·25-s + 0.769·27-s − 1.48·29-s − 0.718·31-s − 0.696·33-s − 1.69·35-s − 0.986·37-s − 0.320·39-s − 2.49·41-s − 0.457·43-s − 2.23·45-s − 0.875·47-s + 3/7·49-s − 2.80·51-s + 0.412·53-s + 1.34·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3732624\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(237.995\)
Root analytic conductor: \(3.92773\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1932} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3732624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good5$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 16 T + 141 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 107 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 5 T + 123 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - T + 73 T^{2} - p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 7 T + 93 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 151 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 129 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 31 T + 417 T^{2} + 31 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 20 T + 289 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
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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

−8.740190511858784941196779843325, −8.579704083504205905857733171891, −8.137659041066516298016612437905, −8.056672261024702803067490546677, −7.43221911252367881547037192202, −7.22186441976559038444930069588, −6.81623291539093492575427722249, −6.66322921388635595777861013257, −5.53016847591230569373791056616, −5.26427202175593157361200938295, −4.78549825442755682913731445418, −4.25570730778494611095714411681, −3.98702310368598651758968513060, −3.73277189873131724902456897366, −2.96681631112852332696080006067, −2.79610498753449416152967632039, −1.81395240462105523158445141432, −1.65579264777971620514258681787, 0, 0, 1.65579264777971620514258681787, 1.81395240462105523158445141432, 2.79610498753449416152967632039, 2.96681631112852332696080006067, 3.73277189873131724902456897366, 3.98702310368598651758968513060, 4.25570730778494611095714411681, 4.78549825442755682913731445418, 5.26427202175593157361200938295, 5.53016847591230569373791056616, 6.66322921388635595777861013257, 6.81623291539093492575427722249, 7.22186441976559038444930069588, 7.43221911252367881547037192202, 8.056672261024702803067490546677, 8.137659041066516298016612437905, 8.579704083504205905857733171891, 8.740190511858784941196779843325

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