Properties

Label 2-2415-1.1-c1-0-62
Degree $2$
Conductor $2415$
Sign $1$
Analytic cond. $19.2838$
Root an. cond. $4.39134$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.72·2-s − 3-s + 5.43·4-s − 5-s − 2.72·6-s + 7-s + 9.37·8-s + 9-s − 2.72·10-s + 2.45·11-s − 5.43·12-s − 0.635·13-s + 2.72·14-s + 15-s + 14.6·16-s − 0.765·17-s + 2.72·18-s + 4.19·19-s − 5.43·20-s − 21-s + 6.69·22-s − 23-s − 9.37·24-s + 25-s − 1.73·26-s − 27-s + 5.43·28-s + ⋯
L(s)  = 1  + 1.92·2-s − 0.577·3-s + 2.71·4-s − 0.447·5-s − 1.11·6-s + 0.377·7-s + 3.31·8-s + 0.333·9-s − 0.862·10-s + 0.740·11-s − 1.56·12-s − 0.176·13-s + 0.728·14-s + 0.258·15-s + 3.67·16-s − 0.185·17-s + 0.642·18-s + 0.963·19-s − 1.21·20-s − 0.218·21-s + 1.42·22-s − 0.208·23-s − 1.91·24-s + 0.200·25-s − 0.339·26-s − 0.192·27-s + 1.02·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(19.2838\)
Root analytic conductor: \(4.39134\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2415,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.544768001\)
\(L(\frac12)\) \(\approx\) \(5.544768001\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 + T \)
good2 \( 1 - 2.72T + 2T^{2} \)
11 \( 1 - 2.45T + 11T^{2} \)
13 \( 1 + 0.635T + 13T^{2} \)
17 \( 1 + 0.765T + 17T^{2} \)
19 \( 1 - 4.19T + 19T^{2} \)
29 \( 1 - 1.83T + 29T^{2} \)
31 \( 1 + 5.20T + 31T^{2} \)
37 \( 1 + 6.88T + 37T^{2} \)
41 \( 1 + 1.85T + 41T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 - 5.66T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 + 4.02T + 59T^{2} \)
61 \( 1 - 5.03T + 61T^{2} \)
67 \( 1 - 8.70T + 67T^{2} \)
71 \( 1 + 9.37T + 71T^{2} \)
73 \( 1 - 1.39T + 73T^{2} \)
79 \( 1 + 7.59T + 79T^{2} \)
83 \( 1 - 17.5T + 83T^{2} \)
89 \( 1 + 7.80T + 89T^{2} \)
97 \( 1 - 6.17T + 97T^{2} \)
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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

−8.875054939410508776071064296124, −7.68124081854851660312407464706, −7.11188331410292626216895228511, −6.45277903154053097083060585353, −5.52715656570914343100389350243, −5.07438513002937791801479627355, −4.11804113842116677795718915829, −3.63063555067820735488778587531, −2.47537721623582506708645740084, −1.32765511466818081337097514675, 1.32765511466818081337097514675, 2.47537721623582506708645740084, 3.63063555067820735488778587531, 4.11804113842116677795718915829, 5.07438513002937791801479627355, 5.52715656570914343100389350243, 6.45277903154053097083060585353, 7.11188331410292626216895228511, 7.68124081854851660312407464706, 8.875054939410508776071064296124

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