Properties

Label 2-15210-1.1-c1-0-29
Degree $2$
Conductor $15210$
Sign $1$
Analytic cond. $121.452$
Root an. cond. $11.0205$
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 + 4-s + 5-s + 3·7-s + 8-s + 10-s + 11-s + 3·14-s + 16-s + 5·19-s + 20-s + 22-s + 4·23-s + 25-s + 3·28-s − 10·31-s + 32-s + 3·35-s + 37-s + 5·38-s + 40-s + 6·41-s − 2·43-s + 44-s + 4·46-s − 9·47-s + 2·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.13·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.801·14-s + 1/4·16-s + 1.14·19-s + 0.223·20-s + 0.213·22-s + 0.834·23-s + 1/5·25-s + 0.566·28-s − 1.79·31-s + 0.176·32-s + 0.507·35-s + 0.164·37-s + 0.811·38-s + 0.158·40-s + 0.937·41-s − 0.304·43-s + 0.150·44-s + 0.589·46-s − 1.31·47-s + 2/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15210\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(121.452\)
Root analytic conductor: \(11.0205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15210,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.270630214\)
\(L(\frac12)\) \(\approx\) \(5.270630214\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−16.15624674257666, −15.17115706453305, −14.88928742667320, −14.30105374819120, −13.96646395983571, −13.23170801369056, −12.83189181584529, −12.08497429099318, −11.54256242309838, −11.06382732749788, −10.62992894545137, −9.682785195732533, −9.298490092241251, −8.467142695680847, −7.882101089271830, −7.179428509343803, −6.738026006344460, −5.764150369247340, −5.308915199241668, −4.853565299372022, −3.990203174962936, −3.352337255502545, −2.439755181311195, −1.717867667751560, −0.9549144894500126, 0.9549144894500126, 1.717867667751560, 2.439755181311195, 3.352337255502545, 3.990203174962936, 4.853565299372022, 5.308915199241668, 5.764150369247340, 6.738026006344460, 7.179428509343803, 7.882101089271830, 8.467142695680847, 9.298490092241251, 9.682785195732533, 10.62992894545137, 11.06382732749788, 11.54256242309838, 12.08497429099318, 12.83189181584529, 13.23170801369056, 13.96646395983571, 14.30105374819120, 14.88928742667320, 15.17115706453305, 16.15624674257666

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