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Properties

Label 2-15210-1.1-c1-0-22
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 + 8-s + 10-s + 16-s − 2·17-s + 8·19-s + 20-s + 4·23-s + 25-s + 2·29-s + 4·31-s + 32-s − 2·34-s − 6·37-s + 8·38-s + 40-s + 10·41-s + 4·46-s + 8·47-s − 7·49-s + 50-s − 6·53-s + 2·58-s + 8·59-s − 2·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1/4·16-s − 0.485·17-s + 1.83·19-s + 0.223·20-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s − 0.986·37-s + 1.29·38-s + 0.158·40-s + 1.56·41-s + 0.589·46-s + 1.16·47-s − 49-s + 0.141·50-s − 0.824·53-s + 0.262·58-s + 1.04·59-s − 0.256·61-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

\(L(1)\) \(\approx\) \(4.427456640\)
\(L(\frac12)\) \(\approx\) \(4.427456640\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−15.97473416091362, −15.53624495981158, −14.81786885409211, −14.19986606641562, −13.93494431928462, −13.15466342970539, −12.95233244983353, −11.96305062245437, −11.79039956508649, −10.99372880285429, −10.46323386497179, −9.826775008236090, −9.180695030787254, −8.673774016033432, −7.667272562654296, −7.327335124207331, −6.543646890088495, −5.978888422729393, −5.281280542294617, −4.799015517797466, −4.033675630750614, −3.127195255568126, −2.709348769586214, −1.681705277309818, −0.8543355977939547, 0.8543355977939547, 1.681705277309818, 2.709348769586214, 3.127195255568126, 4.033675630750614, 4.799015517797466, 5.281280542294617, 5.978888422729393, 6.543646890088495, 7.327335124207331, 7.667272562654296, 8.673774016033432, 9.180695030787254, 9.826775008236090, 10.46323386497179, 10.99372880285429, 11.79039956508649, 11.96305062245437, 12.95233244983353, 13.15466342970539, 13.93494431928462, 14.19986606641562, 14.81786885409211, 15.53624495981158, 15.97473416091362

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