Properties

Label 2-230-1.1-c1-0-7
Degree $2$
Conductor $230$
Sign $1$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.61·3-s + 4-s + 5-s + 1.61·6-s − 0.618·7-s + 8-s − 0.381·9-s + 10-s − 2.85·11-s + 1.61·12-s − 7.09·13-s − 0.618·14-s + 1.61·15-s + 16-s + 6.09·17-s − 0.381·18-s + 1.85·19-s + 20-s − 1.00·21-s − 2.85·22-s + 23-s + 1.61·24-s + 25-s − 7.09·26-s − 5.47·27-s − 0.618·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.934·3-s + 0.5·4-s + 0.447·5-s + 0.660·6-s − 0.233·7-s + 0.353·8-s − 0.127·9-s + 0.316·10-s − 0.860·11-s + 0.467·12-s − 1.96·13-s − 0.165·14-s + 0.417·15-s + 0.250·16-s + 1.47·17-s − 0.0900·18-s + 0.425·19-s + 0.223·20-s − 0.218·21-s − 0.608·22-s + 0.208·23-s + 0.330·24-s + 0.200·25-s − 1.39·26-s − 1.05·27-s − 0.116·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.253074220\)
\(L(\frac12)\) \(\approx\) \(2.253074220\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
23 \( 1 - T \)
good3 \( 1 - 1.61T + 3T^{2} \)
7 \( 1 + 0.618T + 7T^{2} \)
11 \( 1 + 2.85T + 11T^{2} \)
13 \( 1 + 7.09T + 13T^{2} \)
17 \( 1 - 6.09T + 17T^{2} \)
19 \( 1 - 1.85T + 19T^{2} \)
29 \( 1 + 9.23T + 29T^{2} \)
31 \( 1 - 9.09T + 31T^{2} \)
37 \( 1 - 6.47T + 37T^{2} \)
41 \( 1 - 3.32T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 3.70T + 47T^{2} \)
53 \( 1 - 0.472T + 53T^{2} \)
59 \( 1 - 1.70T + 59T^{2} \)
61 \( 1 + 9.32T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 + 4.09T + 71T^{2} \)
73 \( 1 - 3.23T + 73T^{2} \)
79 \( 1 - 1.52T + 79T^{2} \)
83 \( 1 + 6.94T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−12.47883434393512256328344394257, −11.44686251142789410021465049921, −10.01325729367604094589154766938, −9.542030720090398116833551036626, −7.996795640306081590736648146812, −7.37284920887802757557061253147, −5.81865497788505764214963526118, −4.88440935172606354314966556919, −3.21835418156143581887086605491, −2.38964411962337051053951412739, 2.38964411962337051053951412739, 3.21835418156143581887086605491, 4.88440935172606354314966556919, 5.81865497788505764214963526118, 7.37284920887802757557061253147, 7.996795640306081590736648146812, 9.542030720090398116833551036626, 10.01325729367604094589154766938, 11.44686251142789410021465049921, 12.47883434393512256328344394257

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