Properties

Label 2-230-1.1-c5-0-13
Degree $2$
Conductor $230$
Sign $1$
Analytic cond. $36.8882$
Root an. cond. $6.07357$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 10.8·3-s + 16·4-s + 25·5-s − 43.3·6-s + 46.1·7-s − 64·8-s − 125.·9-s − 100·10-s + 8.88·11-s + 173.·12-s + 229.·13-s − 184.·14-s + 270.·15-s + 256·16-s + 561.·17-s + 502.·18-s + 1.95e3·19-s + 400·20-s + 500.·21-s − 35.5·22-s − 529·23-s − 693.·24-s + 625·25-s − 916.·26-s − 3.99e3·27-s + 738.·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.695·3-s + 0.5·4-s + 0.447·5-s − 0.491·6-s + 0.356·7-s − 0.353·8-s − 0.516·9-s − 0.316·10-s + 0.0221·11-s + 0.347·12-s + 0.375·13-s − 0.251·14-s + 0.310·15-s + 0.250·16-s + 0.471·17-s + 0.365·18-s + 1.24·19-s + 0.223·20-s + 0.247·21-s − 0.0156·22-s − 0.208·23-s − 0.245·24-s + 0.200·25-s − 0.265·26-s − 1.05·27-s + 0.178·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(36.8882\)
Root analytic conductor: \(6.07357\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.121997401\)
\(L(\frac12)\) \(\approx\) \(2.121997401\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4T \)
5 \( 1 - 25T \)
23 \( 1 + 529T \)
good3 \( 1 - 10.8T + 243T^{2} \)
7 \( 1 - 46.1T + 1.68e4T^{2} \)
11 \( 1 - 8.88T + 1.61e5T^{2} \)
13 \( 1 - 229.T + 3.71e5T^{2} \)
17 \( 1 - 561.T + 1.41e6T^{2} \)
19 \( 1 - 1.95e3T + 2.47e6T^{2} \)
29 \( 1 - 576.T + 2.05e7T^{2} \)
31 \( 1 - 7.33e3T + 2.86e7T^{2} \)
37 \( 1 - 2.33e3T + 6.93e7T^{2} \)
41 \( 1 - 89.3T + 1.15e8T^{2} \)
43 \( 1 + 1.69e3T + 1.47e8T^{2} \)
47 \( 1 + 2.61e3T + 2.29e8T^{2} \)
53 \( 1 - 1.93e4T + 4.18e8T^{2} \)
59 \( 1 - 3.74e4T + 7.14e8T^{2} \)
61 \( 1 + 1.74e4T + 8.44e8T^{2} \)
67 \( 1 - 2.56e4T + 1.35e9T^{2} \)
71 \( 1 + 6.07e3T + 1.80e9T^{2} \)
73 \( 1 - 6.79e4T + 2.07e9T^{2} \)
79 \( 1 - 7.07e4T + 3.07e9T^{2} \)
83 \( 1 - 5.40e4T + 3.93e9T^{2} \)
89 \( 1 - 6.76e4T + 5.58e9T^{2} \)
97 \( 1 - 1.03e5T + 8.58e9T^{2} \)
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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

−11.26859363965375927121406672334, −10.11556571669804639850258671752, −9.345045317540757415694331994507, −8.400446928609162519051753260858, −7.69781340828240996001091270976, −6.37442454010471592772743557374, −5.23032778226310670134154376878, −3.44455361467418303720275773077, −2.32539462411963892627993392093, −0.972401851147545834387009456483, 0.972401851147545834387009456483, 2.32539462411963892627993392093, 3.44455361467418303720275773077, 5.23032778226310670134154376878, 6.37442454010471592772743557374, 7.69781340828240996001091270976, 8.400446928609162519051753260858, 9.345045317540757415694331994507, 10.11556571669804639850258671752, 11.26859363965375927121406672334

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