Properties

Label 2-7530-1.1-c1-0-107
Degree $2$
Conductor $7530$
Sign $1$
Analytic cond. $60.1273$
Root an. cond. $7.75418$
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 + 7-s + 8-s + 9-s + 10-s − 2·11-s + 12-s + 7·13-s + 14-s + 15-s + 16-s + 4·17-s + 18-s + 6·19-s + 20-s + 21-s − 2·22-s + 3·23-s + 24-s + 25-s + 7·26-s + 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 1.94·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s + 1.37·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7530\)    =    \(2 \cdot 3 \cdot 5 \cdot 251\)
Sign: $1$
Analytic conductor: \(60.1273\)
Root analytic conductor: \(7.75418\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7530,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.636279519\)
\(L(\frac12)\) \(\approx\) \(5.636279519\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
251 \( 1 + T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−7.78459934253036758706951244094, −7.32059077901313203507962473271, −6.30222034688187534190889851491, −5.69464376961664757684368374469, −5.18883416649938731440020864601, −4.23750269616670417315649862823, −3.36842352539423500098910862884, −3.03064552537583404408403798037, −1.78495712120334079063296971883, −1.17910772593497077520236203168, 1.17910772593497077520236203168, 1.78495712120334079063296971883, 3.03064552537583404408403798037, 3.36842352539423500098910862884, 4.23750269616670417315649862823, 5.18883416649938731440020864601, 5.69464376961664757684368374469, 6.30222034688187534190889851491, 7.32059077901313203507962473271, 7.78459934253036758706951244094

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