Properties

Label 2-338130-1.1-c1-0-38
Degree $2$
Conductor $338130$
Sign $-1$
Analytic cond. $2699.98$
Root an. cond. $51.9613$
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  − 2-s + 4-s − 5-s + 2·7-s − 8-s + 10-s − 6·11-s + 13-s − 2·14-s + 16-s − 4·19-s − 20-s + 6·22-s − 4·23-s + 25-s − 26-s + 2·28-s − 8·29-s − 2·31-s − 32-s − 2·35-s + 10·37-s + 4·38-s + 40-s − 6·41-s − 12·43-s − 6·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s − 1.80·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s + 1.27·22-s − 0.834·23-s + 1/5·25-s − 0.196·26-s + 0.377·28-s − 1.48·29-s − 0.359·31-s − 0.176·32-s − 0.338·35-s + 1.64·37-s + 0.648·38-s + 0.158·40-s − 0.937·41-s − 1.82·43-s − 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338130\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2699.98\)
Root analytic conductor: \(51.9613\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{338130} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 338130,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.95676912094155, −12.14582012024206, −11.85324930030028, −11.28350642991532, −10.99107600136011, −10.48072924304352, −10.25307851771132, −9.570993304676830, −9.214602073782164, −8.383839952817447, −8.214228731274741, −7.938968707289875, −7.475642469250516, −6.856106043186881, −6.439967567346017, −5.689899481853863, −5.337256100331559, −4.908066263565162, −4.176889079720633, −3.761387263194363, −3.077939616865447, −2.423403311803040, −2.036820708566472, −1.475655965392391, −0.5366816862542948, 0, 0.5366816862542948, 1.475655965392391, 2.036820708566472, 2.423403311803040, 3.077939616865447, 3.761387263194363, 4.176889079720633, 4.908066263565162, 5.337256100331559, 5.689899481853863, 6.439967567346017, 6.856106043186881, 7.475642469250516, 7.938968707289875, 8.214228731274741, 8.383839952817447, 9.214602073782164, 9.570993304676830, 10.25307851771132, 10.48072924304352, 10.99107600136011, 11.28350642991532, 11.85324930030028, 12.14582012024206, 12.95676912094155

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