Properties

Label 2-23-23.22-c30-0-37
Degree $2$
Conductor $23$
Sign $1$
Analytic cond. $131.132$
Root an. cond. $11.4513$
Motivic weight $30$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.04e4·2-s + 1.97e7·3-s + 1.46e9·4-s − 9.98e11·6-s − 1.98e13·8-s + 1.86e14·9-s + 2.90e16·12-s + 9.69e16·13-s − 5.75e17·16-s − 9.38e18·18-s − 2.66e20·23-s − 3.92e20·24-s + 9.31e20·25-s − 4.88e21·26-s − 3.91e20·27-s − 4.57e21·29-s + 3.15e22·31-s + 5.03e22·32-s + 2.73e23·36-s + 1.91e24·39-s + 3.10e24·41-s + 1.34e25·46-s − 1.77e25·47-s − 1.14e25·48-s + 2.25e25·49-s − 4.69e25·50-s + 1.42e26·52-s + ⋯
L(s)  = 1  − 1.53·2-s + 1.37·3-s + 1.36·4-s − 2.12·6-s − 0.563·8-s + 0.904·9-s + 1.88·12-s + 1.89·13-s − 0.499·16-s − 1.39·18-s − 23-s − 0.777·24-s + 25-s − 2.91·26-s − 0.132·27-s − 0.530·29-s + 1.34·31-s + 1.33·32-s + 1.23·36-s + 2.61·39-s + 1.99·41-s + 1.53·46-s − 1.46·47-s − 0.689·48-s + 49-s − 1.53·50-s + 2.58·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $1$
Analytic conductor: \(131.132\)
Root analytic conductor: \(11.4513\)
Motivic weight: \(30\)
Rational: yes
Arithmetic: yes
Character: $\chi_{23} (22, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :15),\ 1)\)

Particular Values

\(L(\frac{31}{2})\) \(\approx\) \(1.981336616\)
\(L(\frac12)\) \(\approx\) \(1.981336616\)
\(L(16)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + p^{15} T \)
good2 \( 1 + 50407 T + p^{30} T^{2} \)
3 \( 1 - 19799482 T + p^{30} T^{2} \)
5 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
7 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
11 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
13 \( 1 - 96954465195037082 T + p^{30} T^{2} \)
17 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
19 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
29 \( 1 + \)\(45\!\cdots\!74\)\( T + p^{30} T^{2} \)
31 \( 1 - \)\(31\!\cdots\!74\)\( T + p^{30} T^{2} \)
37 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
41 \( 1 - \)\(31\!\cdots\!74\)\( T + p^{30} T^{2} \)
43 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
47 \( 1 + \)\(17\!\cdots\!82\)\( T + p^{30} T^{2} \)
53 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
59 \( 1 - \)\(13\!\cdots\!98\)\( T + p^{30} T^{2} \)
61 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
67 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
71 \( 1 + \)\(32\!\cdots\!26\)\( T + p^{30} T^{2} \)
73 \( 1 + \)\(50\!\cdots\!18\)\( T + p^{30} T^{2} \)
79 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
83 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
89 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
97 \( ( 1 - p^{15} T )( 1 + p^{15} T ) \)
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

−11.32179138845266023916613259846, −10.20636451449380484021649492062, −9.081666179065423840549007780265, −8.464331808217308924425201463959, −7.65872226315270299905087521141, −6.27255152329906035879746329481, −4.04866669702370114220443149468, −2.82950479287640721629202862451, −1.72565852388529464991066952301, −0.795779337645979707375269002458, 0.795779337645979707375269002458, 1.72565852388529464991066952301, 2.82950479287640721629202862451, 4.04866669702370114220443149468, 6.27255152329906035879746329481, 7.65872226315270299905087521141, 8.464331808217308924425201463959, 9.081666179065423840549007780265, 10.20636451449380484021649492062, 11.32179138845266023916613259846

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