Properties

Label 4-1110e2-1.1-c1e2-0-14
Degree $4$
Conductor $1232100$
Sign $1$
Analytic cond. $78.5597$
Root an. cond. $2.97714$
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  − 4-s + 2·5-s − 9-s + 10·11-s + 16-s − 2·20-s − 25-s + 6·29-s + 2·31-s + 36-s − 2·41-s − 10·44-s − 2·45-s + 13·49-s + 20·55-s + 16·59-s + 10·61-s − 64-s − 12·71-s + 2·80-s + 81-s − 16·89-s − 10·99-s + 100-s + 20·101-s − 2·109-s − 6·116-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s − 1/3·9-s + 3.01·11-s + 1/4·16-s − 0.447·20-s − 1/5·25-s + 1.11·29-s + 0.359·31-s + 1/6·36-s − 0.312·41-s − 1.50·44-s − 0.298·45-s + 13/7·49-s + 2.69·55-s + 2.08·59-s + 1.28·61-s − 1/8·64-s − 1.42·71-s + 0.223·80-s + 1/9·81-s − 1.69·89-s − 1.00·99-s + 1/10·100-s + 1.99·101-s − 0.191·109-s − 0.557·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1232100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(78.5597\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1110} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1232100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.008521896\)
\(L(\frac12)\) \(\approx\) \(3.008521896\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
37$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 193 T^{2} + p^{2} T^{4} \)
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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

−9.898806548509306387902157028838, −9.461932428575362119272997614415, −9.432365744164382493535185519442, −8.742533599501486685955799282245, −8.554724625608482015802809910950, −8.346948234449979839643579756319, −7.34736670098169758450464632897, −7.13639854461668622546552540165, −6.50567474779619146881879836596, −6.40974912693474023525917083813, −5.72328564930003353726158881728, −5.60384256760759524094402033659, −4.75956057505067696990028294807, −4.38541092871708392793850025358, −3.78019112165209341356467054099, −3.63997426744686948500345781353, −2.71631083381843678212150511518, −2.12256468552869514027262606471, −1.35581878153588960545147587834, −0.895888343365722567060548398489, 0.895888343365722567060548398489, 1.35581878153588960545147587834, 2.12256468552869514027262606471, 2.71631083381843678212150511518, 3.63997426744686948500345781353, 3.78019112165209341356467054099, 4.38541092871708392793850025358, 4.75956057505067696990028294807, 5.60384256760759524094402033659, 5.72328564930003353726158881728, 6.40974912693474023525917083813, 6.50567474779619146881879836596, 7.13639854461668622546552540165, 7.34736670098169758450464632897, 8.346948234449979839643579756319, 8.554724625608482015802809910950, 8.742533599501486685955799282245, 9.432365744164382493535185519442, 9.461932428575362119272997614415, 9.898806548509306387902157028838

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