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Properties

Label 2-15210-1.1-c1-0-17
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 + 6·11-s + 16-s − 6·17-s + 6·19-s + 20-s − 6·22-s + 6·23-s + 25-s + 6·29-s − 32-s + 6·34-s + 6·37-s − 6·38-s − 40-s − 12·41-s − 8·43-s + 6·44-s − 6·46-s − 7·49-s − 50-s + 12·53-s + 6·55-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.80·11-s + 1/4·16-s − 1.45·17-s + 1.37·19-s + 0.223·20-s − 1.27·22-s + 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.176·32-s + 1.02·34-s + 0.986·37-s − 0.973·38-s − 0.158·40-s − 1.87·41-s − 1.21·43-s + 0.904·44-s − 0.884·46-s − 49-s − 0.141·50-s + 1.64·53-s + 0.809·55-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

\(L(1)\) \(\approx\) \(2.166129572\)
\(L(\frac12)\) \(\approx\) \(2.166129572\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−16.25419612311888, −15.42811057299215, −15.05020315157914, −14.41209751094681, −13.81521191553797, −13.32972108224684, −12.66209138251152, −11.78488001721306, −11.55146159499861, −11.07198153620144, −10.08266088308005, −9.817245909517725, −9.080935894853421, −8.745508129250462, −8.156794110223486, −7.077430178322180, −6.758483153149130, −6.380802747146681, −5.367990359365096, −4.774365788526147, −3.863842276131568, −3.163488915479375, −2.289040260243825, −1.433990240884143, −0.7854370655735896, 0.7854370655735896, 1.433990240884143, 2.289040260243825, 3.163488915479375, 3.863842276131568, 4.774365788526147, 5.367990359365096, 6.380802747146681, 6.758483153149130, 7.077430178322180, 8.156794110223486, 8.745508129250462, 9.080935894853421, 9.817245909517725, 10.08266088308005, 11.07198153620144, 11.55146159499861, 11.78488001721306, 12.66209138251152, 13.32972108224684, 13.81521191553797, 14.41209751094681, 15.05020315157914, 15.42811057299215, 16.25419612311888

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