↑

Properties

Label 1-23-23.7-r1-0-0
Degree $1$
Conductor $23$
Sign $-0.771 - 0.635i$
Analytic cond. $2.47169$
Root an. cond. $2.47169$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.142 − 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (0.654 − 0.755i)5-s + (−0.959 − 0.281i)6-s + (−0.841 + 0.540i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (−0.841 − 0.540i)10-s + (0.142 − 0.989i)11-s + (−0.142 + 0.989i)12-s + (0.841 + 0.540i)13-s + (0.654 + 0.755i)14-s + (−0.415 − 0.909i)15-s + (0.841 − 0.540i)16-s + (0.959 + 0.281i)17-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (0.654 − 0.755i)5-s + (−0.959 − 0.281i)6-s + (−0.841 + 0.540i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (−0.841 − 0.540i)10-s + (0.142 − 0.989i)11-s + (−0.142 + 0.989i)12-s + (0.841 + 0.540i)13-s + (0.654 + 0.755i)14-s + (−0.415 − 0.909i)15-s + (0.841 − 0.540i)16-s + (0.959 + 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.771 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.771 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(23\)
Sign: $-0.771 - 0.635i$
Analytic conductor: \(2.47169\)
Root analytic conductor: \(2.47169\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 23,\ (1:\ ),\ -0.771 - 0.635i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4325295310 - 1.205465147i\)
\(L(\frac12)\) \(\approx\) \(0.4325295310 - 1.205465147i\)
\(L(1)\) \(\approx\) \(0.7210110963 - 0.8129020667i\)
\(L(1)\) \(\approx\) \(0.7210110963 - 0.8129020667i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.142 - 0.989i)T \)
3 \( 1 + (0.415 - 0.909i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (-0.841 + 0.540i)T \)
11 \( 1 + (0.142 - 0.989i)T \)
13 \( 1 + (0.841 + 0.540i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (0.654 + 0.755i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (-0.415 - 0.909i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (-0.959 + 0.281i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (0.654 - 0.755i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−39.07262620557626963054299755984, −37.787570713384329151713533241753, −36.6636214515049842641653433633, −35.29063164677407843707684534234, −33.62823361044807264731901948157, −33.09903504809178900454122103866, −31.95226824792651724429364040440, −30.38871356131503417731294508904, −28.39794589033058331560831436561, −27.0048933421035340969024919826, −25.85554676740181220977013019623, −25.34252927830726541670301819878, −22.98906973737863461487422124845, −22.287419380138063375714722575194, −20.46617546727517632449792563656, −18.675603964503428488534354330079, −17.132093479419417065521898332411, −15.8331749844610280629078898790, −14.590522827823499865063779504670, −13.43210767507056865197446793618, −10.29827245742697055683394829985, −9.45987916574414404551575673409, −7.41010668689748212895182763825, −5.67852031718441859245797855795, −3.60810710722134433432466230901, 1.23377715818156234577714895682, 3.147747137331669402331371101034, 5.88611725344099459803832661243, 8.455756935396261162415773663793, 9.51609515046732405008325293569, 11.76456311027407169240600818462, 13.016062555098127733237425238800, 13.906373355556694967141597978376, 16.63833488600076627074233743463, 18.28452522304774822448264638292, 19.23240608136633369695707634318, 20.56681531641732360817054936241, 21.834517033007257391640124424978, 23.55878638737965249371350251604, 25.08623284191377978574235623390, 26.28902531255279358704820131008, 28.32397460964055898121881676704, 29.08399409536481223018386105355, 30.201494835137184619792322258711, 31.62789800199476039996211751489, 32.456603073918669037352085639277, 35.01877570996695299175752364099, 35.90667704831351102638463650205, 37.005235254945245174840162107254, 37.945948040157155051117051391572

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