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Properties

Label 1-92-92.51-r0-0-0
Degree $1$
Conductor $92$
Sign $0.264 + 0.964i$
Analytic cond. $0.427246$
Root an. cond. $0.427246$
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)3-s + (0.959 + 0.281i)5-s + (−0.654 + 0.755i)7-s + (−0.959 + 0.281i)9-s + (0.841 − 0.540i)11-s + (−0.654 − 0.755i)13-s + (−0.142 + 0.989i)15-s + (−0.415 + 0.909i)17-s + (0.415 + 0.909i)19-s + (−0.841 − 0.540i)21-s + (0.841 + 0.540i)25-s + (−0.415 − 0.909i)27-s + (0.415 − 0.909i)29-s + (0.142 − 0.989i)31-s + (0.654 + 0.755i)33-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)3-s + (0.959 + 0.281i)5-s + (−0.654 + 0.755i)7-s + (−0.959 + 0.281i)9-s + (0.841 − 0.540i)11-s + (−0.654 − 0.755i)13-s + (−0.142 + 0.989i)15-s + (−0.415 + 0.909i)17-s + (0.415 + 0.909i)19-s + (−0.841 − 0.540i)21-s + (0.841 + 0.540i)25-s + (−0.415 − 0.909i)27-s + (0.415 − 0.909i)29-s + (0.142 − 0.989i)31-s + (0.654 + 0.755i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.264 + 0.964i$
Analytic conductor: \(0.427246\)
Root analytic conductor: \(0.427246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 92,\ (0:\ ),\ 0.264 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8699320923 + 0.6631196387i\)
\(L(\frac12)\) \(\approx\) \(0.8699320923 + 0.6631196387i\)
\(L(1)\) \(\approx\) \(1.033054945 + 0.4513325566i\)
\(L(1)\) \(\approx\) \(1.033054945 + 0.4513325566i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.142 + 0.989i)T \)
5 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (-0.654 + 0.755i)T \)
11 \( 1 + (0.841 - 0.540i)T \)
13 \( 1 + (-0.654 - 0.755i)T \)
17 \( 1 + (-0.415 + 0.909i)T \)
19 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (0.142 - 0.989i)T \)
37 \( 1 + (0.959 - 0.281i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 - T \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (0.654 + 0.755i)T \)
61 \( 1 + (0.142 - 0.989i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (0.415 + 0.909i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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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

−29.957098301265712796850910811922, −29.21519580767881848324935665949, −28.4972221313524239598069227621, −26.787330331188542997956348380711, −25.7217595997063384419575631673, −24.93614218484064505911991035725, −23.99120192990708405560022319380, −22.8211134584196502278515789104, −21.78416660011564039529639533407, −20.18844886461354719433391429129, −19.66382137637031813415912478877, −18.20454213289824954877051850700, −17.33970834354132198264668884515, −16.41989369158887725730875839738, −14.45751120981544905280718676820, −13.67420636677125491874065438063, −12.75308002673270881158537386957, −11.56266248497447073972693327360, −9.826931687241675982210444883761, −8.947898974178847682235139484609, −7.0954632579116590441277939116, −6.54783378214076404684605086372, −4.809887759754132055542380554160, −2.80587334312072503139050990222, −1.33413034198152894220738430795, 2.41078521595007270158527949605, 3.685813524694320283470891712127, 5.45702594261120162886315880564, 6.28510851245094473151220477986, 8.41362636400138028612654544132, 9.56279500228719041893709275400, 10.256217758003344275196216268893, 11.70932480926545010870221357393, 13.17535348484071716382517269260, 14.45258856836559178088295450246, 15.282235642973892830609609757470, 16.58379877447502009505215425230, 17.453482373233588609153003344752, 18.90062007297593438612889621206, 20.01618901279982941645308112182, 21.29065221636662376876239679579, 22.08440708297201115061082529859, 22.64430577181990227915578041375, 24.67225826150396045479443123128, 25.38238890610317215086020965424, 26.39715509289183907115402164487, 27.37869859747077600542408831886, 28.52009167964194084292726206820, 29.3378507837610345072127231880, 30.590603920422019803284150590632

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