Properties

Label 2-334620-1.1-c1-0-50
Degree $2$
Conductor $334620$
Sign $-1$
Analytic cond. $2671.95$
Root an. cond. $51.6909$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 3·7-s + 11-s + 4·19-s + 25-s + 6·29-s + 7·31-s − 3·35-s − 2·37-s − 2·41-s + 43-s + 10·47-s + 2·49-s + 6·53-s + 55-s − 6·59-s − 3·61-s + 7·67-s − 4·71-s − 15·73-s − 3·77-s − 5·79-s + 6·83-s − 10·89-s + 4·95-s + 19·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.13·7-s + 0.301·11-s + 0.917·19-s + 1/5·25-s + 1.11·29-s + 1.25·31-s − 0.507·35-s − 0.328·37-s − 0.312·41-s + 0.152·43-s + 1.45·47-s + 2/7·49-s + 0.824·53-s + 0.134·55-s − 0.781·59-s − 0.384·61-s + 0.855·67-s − 0.474·71-s − 1.75·73-s − 0.341·77-s − 0.562·79-s + 0.658·83-s − 1.05·89-s + 0.410·95-s + 1.92·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(334620\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(2671.95\)
Root analytic conductor: \(51.6909\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 334620,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 \)
good7 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.83465348901178, −12.29229714412039, −11.99322906316024, −11.59159409696798, −10.94587443871818, −10.32176080322659, −10.10865168489263, −9.741169659547731, −9.114313039815770, −8.844199097004198, −8.336315945676089, −7.613311764519133, −7.262002987294729, −6.671133741247666, −6.328164218538346, −5.871778059534273, −5.401511812413671, −4.728670810783226, −4.323894852190494, −3.611603196954764, −3.141345543337913, −2.710948351687518, −2.154386187173134, −1.260939899069805, −0.8623074149573226, 0, 0.8623074149573226, 1.260939899069805, 2.154386187173134, 2.710948351687518, 3.141345543337913, 3.611603196954764, 4.323894852190494, 4.728670810783226, 5.401511812413671, 5.871778059534273, 6.328164218538346, 6.671133741247666, 7.262002987294729, 7.613311764519133, 8.336315945676089, 8.844199097004198, 9.114313039815770, 9.741169659547731, 10.10865168489263, 10.32176080322659, 10.94587443871818, 11.59159409696798, 11.99322906316024, 12.29229714412039, 12.83465348901178

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