Properties

Label 2-3-1.1-c23-0-2
Degree $2$
Conductor $3$
Sign $-1$
Analytic cond. $10.0561$
Root an. cond. $3.17113$
Motivic weight $23$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.12e3·2-s + 1.77e5·3-s − 7.11e6·4-s − 4.88e7·5-s + 1.99e8·6-s − 1.72e9·7-s − 1.74e10·8-s + 3.13e10·9-s − 5.51e10·10-s − 1.42e12·11-s − 1.26e12·12-s − 8.22e12·13-s − 1.94e12·14-s − 8.65e12·15-s + 3.99e13·16-s − 5.98e12·17-s + 3.53e13·18-s + 6.80e14·19-s + 3.47e14·20-s − 3.05e14·21-s − 1.61e15·22-s + 1.54e13·23-s − 3.09e15·24-s − 9.53e15·25-s − 9.27e15·26-s + 5.55e15·27-s + 1.22e16·28-s + ⋯
L(s)  = 1  + 0.389·2-s + 0.577·3-s − 0.848·4-s − 0.447·5-s + 0.224·6-s − 0.329·7-s − 0.719·8-s + 1/3·9-s − 0.174·10-s − 1.50·11-s − 0.489·12-s − 1.27·13-s − 0.128·14-s − 0.258·15-s + 0.567·16-s − 0.0423·17-s + 0.129·18-s + 1.33·19-s + 0.379·20-s − 0.190·21-s − 0.587·22-s + 0.00337·23-s − 0.415·24-s − 0.799·25-s − 0.495·26-s + 0.192·27-s + 0.279·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-1$
Analytic conductor: \(10.0561\)
Root analytic conductor: \(3.17113\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3,\ (\ :23/2),\ -1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p^{11} T \)
good2 \( 1 - 141 p^{3} T + p^{23} T^{2} \)
5 \( 1 + 9772746 p T + p^{23} T^{2} \)
7 \( 1 + 35177320 p^{2} T + p^{23} T^{2} \)
11 \( 1 + 129842107284 p T + p^{23} T^{2} \)
13 \( 1 + 632381849602 p T + p^{23} T^{2} \)
17 \( 1 + 5989210330446 T + p^{23} T^{2} \)
19 \( 1 - 35789762172404 p T + p^{23} T^{2} \)
23 \( 1 - 15440648191080 T + p^{23} T^{2} \)
29 \( 1 - 115094192813324022 T + p^{23} T^{2} \)
31 \( 1 + 90829724501108800 T + p^{23} T^{2} \)
37 \( 1 + 1297873386623227570 T + p^{23} T^{2} \)
41 \( 1 - 5214036225478655130 T + p^{23} T^{2} \)
43 \( 1 + 2410434516296794108 T + p^{23} T^{2} \)
47 \( 1 + 23132669525900803824 T + p^{23} T^{2} \)
53 \( 1 + 44512631945276522850 T + p^{23} T^{2} \)
59 \( 1 + \)\(32\!\cdots\!76\)\( T + p^{23} T^{2} \)
61 \( 1 + \)\(19\!\cdots\!22\)\( T + p^{23} T^{2} \)
67 \( 1 + \)\(64\!\cdots\!96\)\( T + p^{23} T^{2} \)
71 \( 1 - \)\(35\!\cdots\!12\)\( T + p^{23} T^{2} \)
73 \( 1 - \)\(33\!\cdots\!70\)\( T + p^{23} T^{2} \)
79 \( 1 + \)\(68\!\cdots\!20\)\( T + p^{23} T^{2} \)
83 \( 1 + \)\(11\!\cdots\!44\)\( T + p^{23} T^{2} \)
89 \( 1 + \)\(23\!\cdots\!74\)\( T + p^{23} T^{2} \)
97 \( 1 + \)\(30\!\cdots\!86\)\( T + p^{23} T^{2} \)
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

−19.55848746890403235781033512048, −18.03792405101557126482274858790, −15.65349536356501171189382382218, −14.00196899466533255903183083900, −12.53013863163623625579100748314, −9.770527041793505405321287691707, −7.85748098486953502191745876535, −4.95545867931327919581802415184, −3.04577515234153929617839320136, 0, 3.04577515234153929617839320136, 4.95545867931327919581802415184, 7.85748098486953502191745876535, 9.770527041793505405321287691707, 12.53013863163623625579100748314, 14.00196899466533255903183083900, 15.65349536356501171189382382218, 18.03792405101557126482274858790, 19.55848746890403235781033512048

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