Properties

Label 2-230-1.1-c1-0-4
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 − 0.618·3-s + 4-s + 5-s − 0.618·6-s + 1.61·7-s + 8-s − 2.61·9-s + 10-s + 3.85·11-s − 0.618·12-s + 4.09·13-s + 1.61·14-s − 0.618·15-s + 16-s − 5.09·17-s − 2.61·18-s − 4.85·19-s + 20-s − 1.00·21-s + 3.85·22-s + 23-s − 0.618·24-s + 25-s + 4.09·26-s + 3.47·27-s + 1.61·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.356·3-s + 0.5·4-s + 0.447·5-s − 0.252·6-s + 0.611·7-s + 0.353·8-s − 0.872·9-s + 0.316·10-s + 1.16·11-s − 0.178·12-s + 1.13·13-s + 0.432·14-s − 0.159·15-s + 0.250·16-s − 1.23·17-s − 0.617·18-s − 1.11·19-s + 0.223·20-s − 0.218·21-s + 0.821·22-s + 0.208·23-s − 0.126·24-s + 0.200·25-s + 0.802·26-s + 0.668·27-s + 0.305·28-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

\(L(1)\) \(\approx\) \(1.801075793\)
\(L(\frac12)\) \(\approx\) \(1.801075793\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618T + 3T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 - 3.85T + 11T^{2} \)
13 \( 1 - 4.09T + 13T^{2} \)
17 \( 1 + 5.09T + 17T^{2} \)
19 \( 1 + 4.85T + 19T^{2} \)
29 \( 1 + 4.76T + 29T^{2} \)
31 \( 1 + 2.09T + 31T^{2} \)
37 \( 1 + 2.47T + 37T^{2} \)
41 \( 1 + 12.3T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 9.70T + 47T^{2} \)
53 \( 1 + 8.47T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 - 6.32T + 61T^{2} \)
67 \( 1 - 5.52T + 67T^{2} \)
71 \( 1 - 7.09T + 71T^{2} \)
73 \( 1 + 1.23T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + 1.52T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−12.15772750434465927058119918655, −11.19558779221915460508966203868, −10.82084895697821165088256119833, −9.140306759551471116968745802587, −8.377340443911295124746456879666, −6.70407123397807318145608994971, −6.06854457591781683225897468344, −4.89069048655957089784278056400, −3.67058546540356043216825506916, −1.89317783414266679508333114377, 1.89317783414266679508333114377, 3.67058546540356043216825506916, 4.89069048655957089784278056400, 6.06854457591781683225897468344, 6.70407123397807318145608994971, 8.377340443911295124746456879666, 9.140306759551471116968745802587, 10.82084895697821165088256119833, 11.19558779221915460508966203868, 12.15772750434465927058119918655

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