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Properties

Label 2-92-23.2-c1-0-0
Degree $2$
Conductor $92$
Sign $0.729 - 0.684i$
Analytic cond. $0.734623$
Root an. cond. $0.857101$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.210 + 1.46i)3-s + (0.926 + 0.272i)5-s + (−1.14 + 1.32i)7-s + (0.785 − 0.230i)9-s + (0.436 − 0.280i)11-s + (−2.04 − 2.35i)13-s + (−0.203 + 1.41i)15-s + (1.39 − 3.05i)17-s + (−2.90 − 6.36i)19-s + (−2.17 − 1.40i)21-s + (−1.35 + 4.60i)23-s + (−3.42 − 2.19i)25-s + (2.34 + 5.13i)27-s + (1.01 − 2.21i)29-s + (−0.253 + 1.76i)31-s + ⋯
L(s)  = 1  + (0.121 + 0.843i)3-s + (0.414 + 0.121i)5-s + (−0.434 + 0.501i)7-s + (0.261 − 0.0768i)9-s + (0.131 − 0.0845i)11-s + (−0.566 − 0.654i)13-s + (−0.0524 + 0.364i)15-s + (0.338 − 0.740i)17-s + (−0.666 − 1.46i)19-s + (−0.475 − 0.305i)21-s + (−0.282 + 0.959i)23-s + (−0.684 − 0.439i)25-s + (0.450 + 0.987i)27-s + (0.188 − 0.412i)29-s + (−0.0454 + 0.316i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.729 - 0.684i$
Analytic conductor: \(0.734623\)
Root analytic conductor: \(0.857101\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 92,\ (\ :1/2),\ 0.729 - 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.974050 + 0.385623i\)
\(L(\frac12)\) \(\approx\) \(0.974050 + 0.385623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + (1.35 - 4.60i)T \)
good3 \( 1 + (-0.210 - 1.46i)T + (-2.87 + 0.845i)T^{2} \)
5 \( 1 + (-0.926 - 0.272i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (1.14 - 1.32i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (-0.436 + 0.280i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (2.04 + 2.35i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-1.39 + 3.05i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (2.90 + 6.36i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (-1.01 + 2.21i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.253 - 1.76i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (4.58 - 1.34i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (-12.0 - 3.54i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (0.236 + 1.64i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 - 5.46T + 47T^{2} \)
53 \( 1 + (8.08 - 9.32i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (0.434 + 0.500i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (0.685 - 4.76i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (9.51 + 6.11i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (4.30 + 2.76i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-5.40 - 11.8i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (9.56 + 11.0i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (-2.91 + 0.854i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (1.78 + 12.4i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (0.645 + 0.189i)T + (81.6 + 52.4i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.29643599728722020389167437357, −13.17792279286181374605871329143, −12.09424949287418530683732839216, −10.76083190376504744449616307848, −9.738496449049352445334333549252, −9.073880864857998544431910803267, −7.38914735333399448557058400501, −5.90621966047291652155226060729, −4.55649725117800984101633133659, −2.87271157771041116109145278792, 1.88651142605817688110252411733, 4.10055755733027104876728142176, 6.01656916805630417788144171389, 7.07764940851475640223032654103, 8.179486869148536406465343017569, 9.655645674845707814589151454928, 10.57663904045988798365334061599, 12.23733338297074728087565847737, 12.79064336001435995505304768343, 13.87746279969656257270825461503

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