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Properties

Label 2-15210-1.1-c1-0-28
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 + 4·7-s + 8-s − 10-s + 4·14-s + 16-s + 6·17-s + 4·19-s − 20-s − 6·23-s + 25-s + 4·28-s + 6·29-s + 10·31-s + 32-s + 6·34-s − 4·35-s + 10·37-s + 4·38-s − 40-s − 6·41-s − 4·43-s − 6·46-s + 12·47-s + 9·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.755·28-s + 1.11·29-s + 1.79·31-s + 0.176·32-s + 1.02·34-s − 0.676·35-s + 1.64·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.884·46-s + 1.75·47-s + 9/7·49-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

\(L(1)\) \(\approx\) \(4.780265780\)
\(L(\frac12)\) \(\approx\) \(4.780265780\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.90013571710671, −15.32765013701608, −14.95068734295928, −14.21014348361752, −13.91550171474862, −13.56241943296718, −12.37914217938914, −12.02912469192993, −11.84424678186356, −11.07217948615906, −10.50545147691836, −9.936468112331188, −9.177792513841780, −8.177518271705935, −7.833356831161605, −7.638349180793635, −6.482027009558117, −5.999514110451615, −5.140314743416681, −4.729881922258882, −4.140204268596983, −3.269739405195133, −2.626310292152584, −1.574489412847685, −0.9382379566034233, 0.9382379566034233, 1.574489412847685, 2.626310292152584, 3.269739405195133, 4.140204268596983, 4.729881922258882, 5.140314743416681, 5.999514110451615, 6.482027009558117, 7.638349180793635, 7.833356831161605, 8.177518271705935, 9.177792513841780, 9.936468112331188, 10.50545147691836, 11.07217948615906, 11.84424678186356, 12.02912469192993, 12.37914217938914, 13.56241943296718, 13.91550171474862, 14.21014348361752, 14.95068734295928, 15.32765013701608, 15.90013571710671

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