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Properties

Label 1-23-23.13-r0-0-0
Degree $1$
Conductor $23$
Sign $0.0117 + 0.999i$
Analytic cond. $0.106811$
Root an. cond. $0.106811$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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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

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0117 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0117 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(23\)
Sign: $0.0117 + 0.999i$
Analytic conductor: \(0.106811\)
Root analytic conductor: \(0.106811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 23,\ (0:\ ),\ 0.0117 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4542820890 + 0.4596636635i\)
\(L(\frac12)\) \(\approx\) \(0.4542820890 + 0.4596636635i\)
\(L(1)\) \(\approx\) \(0.7124830147 + 0.4910059593i\)
\(L(1)\) \(\approx\) \(0.7124830147 + 0.4910059593i\)

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 \( 1 + (-0.654 - 0.755i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−38.50340307558800926550590898790, −37.5173950391724365292793859954, −36.23026982523501020290933788759, −35.405010925966572274206402222, −33.71450450824251765414541776951, −31.51461797195130987311578820832, −30.65281967237935792977330619243, −30.00548137405077635256440342745, −28.42778028807417555980305445107, −26.96277899701958235711638522869, −25.83869469482731532443275203768, −23.75888055708372168717217522032, −22.84970431975498903297629724134, −20.87208035387172726679522644384, −19.77526761173815985849828230239, −18.55761841830637896742463096719, −17.5871153342747464320750956621, −14.77864637462672441075329062038, −13.52059537639149946355024865231, −11.94789351008338603970116213884, −10.75645338483036295494981316768, −8.580891341384805742436590726156, −7.178383719762419099693442656676, −4.03528117503497395606199794164, −2.044762235917227441005602451891, 4.12168134698006821351736110537, 5.527258524941725784466998943930, 8.23455744973127504023626840065, 8.828560229821020532492420431612, 11.00692064811433854564371405694, 13.380213450669045687542426834143, 15.04251051590667695399815539530, 15.85253243312657114363761589162, 17.16891944881079233650210363882, 19.00505634356831335793236361495, 20.67494430032699797550929546084, 22.066231640167673219479742483795, 23.76188687440743738536230743016, 24.83033504983248348613401784470, 26.275631990238364246164021111486, 27.50777201752062610708812593831, 28.1210607618274055065119894514, 30.97805367598638152476017439155, 31.87412179805184831834560608843, 32.87424424000170917884878324688, 34.187775530292962937658311082520, 35.31986669431009589084467665916, 36.83136652728095244607274574675, 37.77411630115861677116390700405, 39.630589253427970962995821234711

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