Properties

Label 2-338130-1.1-c1-0-16
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 4·7-s − 8-s + 10-s + 3·11-s + 13-s + 4·14-s + 16-s + 5·19-s − 20-s − 3·22-s − 4·23-s + 25-s − 26-s − 4·28-s + 10·29-s + 31-s − 32-s + 4·35-s + 37-s − 5·38-s + 40-s − 6·41-s − 9·43-s + 3·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s + 0.904·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.14·19-s − 0.223·20-s − 0.639·22-s − 0.834·23-s + 1/5·25-s − 0.196·26-s − 0.755·28-s + 1.85·29-s + 0.179·31-s − 0.176·32-s + 0.676·35-s + 0.164·37-s − 0.811·38-s + 0.158·40-s − 0.937·41-s − 1.37·43-s + 0.452·44-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: \(0\)
Selberg data: \((2,\ 338130,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9890670549\)
\(L(\frac12)\) \(\approx\) \(0.9890670549\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
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

−12.19036500710231, −12.04698031468380, −11.89978555205292, −11.23984062673862, −10.61493490910058, −10.14466807392607, −9.836613830384686, −9.498261144508115, −8.930522208775651, −8.470411016287466, −8.111212689595031, −7.528260767842494, −6.813022463627655, −6.675924513149005, −6.354480337090747, −5.624101606708627, −5.157206469354928, −4.386277844981672, −3.797004827648581, −3.430680648122585, −2.934757954649311, −2.411032115964726, −1.511928443423033, −1.005474358375783, −0.3422091437740553, 0.3422091437740553, 1.005474358375783, 1.511928443423033, 2.411032115964726, 2.934757954649311, 3.430680648122585, 3.797004827648581, 4.386277844981672, 5.157206469354928, 5.624101606708627, 6.354480337090747, 6.675924513149005, 6.813022463627655, 7.528260767842494, 8.111212689595031, 8.470411016287466, 8.930522208775651, 9.498261144508115, 9.836613830384686, 10.14466807392607, 10.61493490910058, 11.23984062673862, 11.89978555205292, 12.04698031468380, 12.19036500710231

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