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Properties

Label 2-334620-1.1-c1-0-44
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 + 7-s − 11-s − 6·17-s + 6·19-s + 23-s + 25-s − 7·29-s − 31-s + 35-s − 10·37-s + 9·41-s + 43-s − 47-s − 6·49-s − 7·53-s − 55-s + 6·59-s + 2·61-s + 4·67-s − 6·71-s − 73-s − 77-s − 8·79-s − 4·83-s − 6·85-s + 14·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.301·11-s − 1.45·17-s + 1.37·19-s + 0.208·23-s + 1/5·25-s − 1.29·29-s − 0.179·31-s + 0.169·35-s − 1.64·37-s + 1.40·41-s + 0.152·43-s − 0.145·47-s − 6/7·49-s − 0.961·53-s − 0.134·55-s + 0.781·59-s + 0.256·61-s + 0.488·67-s − 0.712·71-s − 0.117·73-s − 0.113·77-s − 0.900·79-s − 0.439·83-s − 0.650·85-s + 1.48·89-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 \( 1 - T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−12.93319768688771, −12.42640260024608, −11.82544125827394, −11.27446056395279, −11.17958581509185, −10.58718847941766, −10.03548424257107, −9.664204487443716, −9.007542479002844, −8.907514221785525, −8.272471923564658, −7.585964892844846, −7.355229219613698, −6.858838424045301, −6.193986216803974, −5.832565347390828, −5.221901966703194, −4.861318898225581, −4.392966998769324, −3.590236549465063, −3.291935255631525, −2.500027631124270, −2.025963131645738, −1.548055436752564, −0.7675957093317002, 0, 0.7675957093317002, 1.548055436752564, 2.025963131645738, 2.500027631124270, 3.291935255631525, 3.590236549465063, 4.392966998769324, 4.861318898225581, 5.221901966703194, 5.832565347390828, 6.193986216803974, 6.858838424045301, 7.355229219613698, 7.585964892844846, 8.272471923564658, 8.907514221785525, 9.007542479002844, 9.664204487443716, 10.03548424257107, 10.58718847941766, 11.17958581509185, 11.27446056395279, 11.82544125827394, 12.42640260024608, 12.93319768688771

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