Properties

Label 2-374790-1.1-c1-0-24
Degree $2$
Conductor $374790$
Sign $1$
Analytic cond. $2992.71$
Root an. cond. $54.7056$
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 + 3-s + 4-s + 5-s − 6-s − 2·7-s − 8-s + 9-s − 10-s − 5·11-s + 12-s + 13-s + 2·14-s + 15-s + 16-s − 7·17-s − 18-s + 20-s − 2·21-s + 5·22-s + 3·23-s − 24-s + 25-s − 26-s + 27-s − 2·28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s + 0.288·12-s + 0.277·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 1.69·17-s − 0.235·18-s + 0.223·20-s − 0.436·21-s + 1.06·22-s + 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.377·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(374790\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(2992.71\)
Root analytic conductor: \(54.7056\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 374790,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.418888832\)
\(L(\frac12)\) \(\approx\) \(2.418888832\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 - T \)
31 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−12.62075987766701, −12.11240283803357, −11.43594947245115, −10.85086633421344, −10.64966699575958, −10.23062905164899, −9.733306003383599, −9.265841896936243, −8.848860586699332, −8.490736151849587, −8.097240187873782, −7.308389573746952, −7.172035231586068, −6.560747145867678, −6.161198772324704, −5.510845926085967, −5.115570912080990, −4.386651115504842, −3.942183546979691, −3.151236193664942, −2.744225613693959, −2.263021166999354, −1.987117678273550, −0.8213454052628998, −0.5539783534487513, 0.5539783534487513, 0.8213454052628998, 1.987117678273550, 2.263021166999354, 2.744225613693959, 3.151236193664942, 3.942183546979691, 4.386651115504842, 5.115570912080990, 5.510845926085967, 6.161198772324704, 6.560747145867678, 7.172035231586068, 7.308389573746952, 8.097240187873782, 8.490736151849587, 8.848860586699332, 9.265841896936243, 9.733306003383599, 10.23062905164899, 10.64966699575958, 10.85086633421344, 11.43594947245115, 12.11240283803357, 12.62075987766701

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