Properties

Label 2-25050-1.1-c1-0-16
Degree $2$
Conductor $25050$
Sign $1$
Analytic cond. $200.025$
Root an. cond. $14.1430$
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 + 6-s + 4·7-s + 8-s + 9-s + 5·11-s + 12-s − 3·13-s + 4·14-s + 16-s + 7·17-s + 18-s − 3·19-s + 4·21-s + 5·22-s + 7·23-s + 24-s − 3·26-s + 27-s + 4·28-s − 9·29-s + 7·31-s + 32-s + 5·33-s + 7·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.50·11-s + 0.288·12-s − 0.832·13-s + 1.06·14-s + 1/4·16-s + 1.69·17-s + 0.235·18-s − 0.688·19-s + 0.872·21-s + 1.06·22-s + 1.45·23-s + 0.204·24-s − 0.588·26-s + 0.192·27-s + 0.755·28-s − 1.67·29-s + 1.25·31-s + 0.176·32-s + 0.870·33-s + 1.20·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 167\)
Sign: $1$
Analytic conductor: \(200.025\)
Root analytic conductor: \(14.1430\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 25050,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.496219557\)
\(L(\frac12)\) \(\approx\) \(7.496219557\)
\(L(\frac{3}{2})\) 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 \( 1 \)
167 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.93906176476402, −14.80246924734818, −14.45083028018454, −13.88682256360882, −13.41812586859252, −12.55939699421256, −12.16696763074054, −11.71665302104645, −11.17298403182812, −10.56037733401418, −9.951430643612505, −9.145815526321721, −8.849234648054330, −7.973953562148219, −7.580468878610525, −7.065937184851148, −6.340098281726479, −5.526397731152676, −5.034519471185700, −4.392311984132340, −3.851874430037059, −3.149543745936559, −2.342590356688138, −1.563696377159188, −1.059424648299673, 1.059424648299673, 1.563696377159188, 2.342590356688138, 3.149543745936559, 3.851874430037059, 4.392311984132340, 5.034519471185700, 5.526397731152676, 6.340098281726479, 7.065937184851148, 7.580468878610525, 7.973953562148219, 8.849234648054330, 9.145815526321721, 9.951430643612505, 10.56037733401418, 11.17298403182812, 11.71665302104645, 12.16696763074054, 12.55939699421256, 13.41812586859252, 13.88682256360882, 14.45083028018454, 14.80246924734818, 14.93906176476402

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