Properties

Label 2-230-1.1-c5-0-7
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 0.233·3-s + 16·4-s − 25·5-s − 0.935·6-s + 203.·7-s − 64·8-s − 242.·9-s + 100·10-s − 151.·11-s + 3.74·12-s + 1.15e3·13-s − 814.·14-s − 5.84·15-s + 256·16-s − 692.·17-s + 971.·18-s − 360.·19-s − 400·20-s + 47.6·21-s + 604.·22-s + 529·23-s − 14.9·24-s + 625·25-s − 4.61e3·26-s − 113.·27-s + 3.25e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.0150·3-s + 0.5·4-s − 0.447·5-s − 0.0106·6-s + 1.56·7-s − 0.353·8-s − 0.999·9-s + 0.316·10-s − 0.376·11-s + 0.00750·12-s + 1.89·13-s − 1.11·14-s − 0.00671·15-s + 0.250·16-s − 0.581·17-s + 0.706·18-s − 0.229·19-s − 0.223·20-s + 0.0235·21-s + 0.266·22-s + 0.208·23-s − 0.00530·24-s + 0.200·25-s − 1.33·26-s − 0.0300·27-s + 0.784·28-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.484395079\)
\(L(\frac12)\) \(\approx\) \(1.484395079\)
\(L(\frac{7}{2})\) 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 \( 1 - 0.233T + 243T^{2} \)
7 \( 1 - 203.T + 1.68e4T^{2} \)
11 \( 1 + 151.T + 1.61e5T^{2} \)
13 \( 1 - 1.15e3T + 3.71e5T^{2} \)
17 \( 1 + 692.T + 1.41e6T^{2} \)
19 \( 1 + 360.T + 2.47e6T^{2} \)
29 \( 1 + 3.68e3T + 2.05e7T^{2} \)
31 \( 1 - 868.T + 2.86e7T^{2} \)
37 \( 1 - 4.02e3T + 6.93e7T^{2} \)
41 \( 1 + 6.44e3T + 1.15e8T^{2} \)
43 \( 1 - 6.56e3T + 1.47e8T^{2} \)
47 \( 1 - 1.28e4T + 2.29e8T^{2} \)
53 \( 1 + 1.15e4T + 4.18e8T^{2} \)
59 \( 1 - 2.86e4T + 7.14e8T^{2} \)
61 \( 1 - 3.67e3T + 8.44e8T^{2} \)
67 \( 1 - 3.94e4T + 1.35e9T^{2} \)
71 \( 1 - 3.18e4T + 1.80e9T^{2} \)
73 \( 1 - 2.91e3T + 2.07e9T^{2} \)
79 \( 1 - 1.00e5T + 3.07e9T^{2} \)
83 \( 1 - 7.71e4T + 3.93e9T^{2} \)
89 \( 1 - 1.06e5T + 5.58e9T^{2} \)
97 \( 1 + 3.58e4T + 8.58e9T^{2} \)
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

−11.09388844430546426587136207170, −10.79099204512765737190926797396, −9.026577096752565762445732440820, −8.382831298562673535696631739702, −7.75658069396741811043955260953, −6.31234196687001658927626908922, −5.17362544898285878903348890785, −3.73706323688091016719729290349, −2.14777927476108878715770390398, −0.827117208378010247823547064852, 0.827117208378010247823547064852, 2.14777927476108878715770390398, 3.73706323688091016719729290349, 5.17362544898285878903348890785, 6.31234196687001658927626908922, 7.75658069396741811043955260953, 8.382831298562673535696631739702, 9.026577096752565762445732440820, 10.79099204512765737190926797396, 11.09388844430546426587136207170

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