Properties

Degree $2$
Conductor $25230$
Sign $1$
Motivic weight $1$
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 − 4·7-s + 8-s + 9-s − 10-s − 12-s + 2·13-s − 4·14-s + 15-s + 16-s − 6·17-s + 18-s + 4·19-s − 20-s + 4·21-s − 24-s + 25-s + 2·26-s − 27-s − 4·28-s + 30-s − 8·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.554·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.872·21-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.755·28-s + 0.182·30-s − 1.43·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25230\)    =    \(2 \cdot 3 \cdot 5 \cdot 29^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{25230} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 25230,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.244882331\)
\(L(\frac12)\) \(\approx\) \(1.244882331\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
29 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.50531180012370, −14.94617593639065, −14.16739882218056, −13.64343947988588, −13.04952451960521, −12.77167474994547, −12.27602467989280, −11.55667746901086, −11.12637348740041, −10.67474276603802, −9.977715282416368, −9.295317969736511, −8.950323479234566, −7.990195365669600, −7.296445249901619, −6.761574656261500, −6.406658149129514, −5.664008154924736, −5.242772249443074, −4.205249492591677, −3.944642843606015, −3.156187812548260, −2.560029928607487, −1.503189827464697, −0.4044833625699468, 0.4044833625699468, 1.503189827464697, 2.560029928607487, 3.156187812548260, 3.944642843606015, 4.205249492591677, 5.242772249443074, 5.664008154924736, 6.406658149129514, 6.761574656261500, 7.296445249901619, 7.990195365669600, 8.950323479234566, 9.295317969736511, 9.977715282416368, 10.67474276603802, 11.12637348740041, 11.55667746901086, 12.27602467989280, 12.77167474994547, 13.04952451960521, 13.64343947988588, 14.16739882218056, 14.94617593639065, 15.50531180012370

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