Properties

Label 2-230-1.1-c1-0-6
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.43·3-s + 4-s − 5-s + 1.43·6-s + 3.08·7-s + 8-s − 0.950·9-s − 10-s − 6.46·11-s + 1.43·12-s + 3.95·13-s + 3.08·14-s − 1.43·15-s + 16-s − 3.43·17-s − 0.950·18-s + 3.08·19-s − 20-s + 4.41·21-s − 6.46·22-s − 23-s + 1.43·24-s + 25-s + 3.95·26-s − 5.65·27-s + 3.08·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.826·3-s + 0.5·4-s − 0.447·5-s + 0.584·6-s + 1.16·7-s + 0.353·8-s − 0.316·9-s − 0.316·10-s − 1.95·11-s + 0.413·12-s + 1.09·13-s + 0.825·14-s − 0.369·15-s + 0.250·16-s − 0.832·17-s − 0.224·18-s + 0.708·19-s − 0.223·20-s + 0.964·21-s − 1.37·22-s − 0.208·23-s + 0.292·24-s + 0.200·25-s + 0.774·26-s − 1.08·27-s + 0.583·28-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

\(L(1)\) \(\approx\) \(2.160604022\)
\(L(\frac12)\) \(\approx\) \(2.160604022\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.43T + 3T^{2} \)
7 \( 1 - 3.08T + 7T^{2} \)
11 \( 1 + 6.46T + 11T^{2} \)
13 \( 1 - 3.95T + 13T^{2} \)
17 \( 1 + 3.43T + 17T^{2} \)
19 \( 1 - 3.08T + 19T^{2} \)
29 \( 1 - 0.863T + 29T^{2} \)
31 \( 1 + 5.95T + 31T^{2} \)
37 \( 1 + 7.03T + 37T^{2} \)
41 \( 1 - 5.60T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 3.90T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 6.86T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 - 2.56T + 71T^{2} \)
73 \( 1 - 5.90T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 - 9.03T + 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.33475975770652326267792039030, −11.09848412659422462184542358089, −10.76551686819435132078406800875, −9.000035973415752322031265082013, −8.063154177754476700386349911353, −7.52358087496945188915114426824, −5.76389467487929549441661298382, −4.78597473354651412117779600274, −3.44779765652501183131218920925, −2.22861792377438448870645236794, 2.22861792377438448870645236794, 3.44779765652501183131218920925, 4.78597473354651412117779600274, 5.76389467487929549441661298382, 7.52358087496945188915114426824, 8.063154177754476700386349911353, 9.000035973415752322031265082013, 10.76551686819435132078406800875, 11.09848412659422462184542358089, 12.33475975770652326267792039030

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