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Properties

Label 2-92-23.6-c1-0-1
Degree $2$
Conductor $92$
Sign $0.636 + 0.770i$
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  + (−1.96 − 0.576i)3-s + (3.22 − 2.07i)5-s + (0.267 − 1.86i)7-s + (1.00 + 0.644i)9-s + (0.804 + 1.76i)11-s + (−0.112 − 0.780i)13-s + (−7.53 + 2.21i)15-s + (−4.90 + 5.66i)17-s + (3.96 + 4.57i)19-s + (−1.60 + 3.50i)21-s + (−1.77 − 4.45i)23-s + (4.03 − 8.84i)25-s + (2.42 + 2.79i)27-s + (−1.75 + 2.03i)29-s + (2.65 − 0.780i)31-s + ⋯
L(s)  = 1  + (−1.13 − 0.333i)3-s + (1.44 − 0.927i)5-s + (0.101 − 0.704i)7-s + (0.334 + 0.214i)9-s + (0.242 + 0.531i)11-s + (−0.0311 − 0.216i)13-s + (−1.94 + 0.571i)15-s + (−1.18 + 1.37i)17-s + (0.909 + 1.04i)19-s + (−0.349 + 0.764i)21-s + (−0.370 − 0.928i)23-s + (0.807 − 1.76i)25-s + (0.466 + 0.538i)27-s + (−0.326 + 0.377i)29-s + (0.477 − 0.140i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.636 + 0.770i$
Analytic conductor: \(0.734623\)
Root analytic conductor: \(0.857101\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 92,\ (\ :1/2),\ 0.636 + 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.775831 - 0.365432i\)
\(L(\frac12)\) \(\approx\) \(0.775831 - 0.365432i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.77 + 4.45i)T \)
good3 \( 1 + (1.96 + 0.576i)T + (2.52 + 1.62i)T^{2} \)
5 \( 1 + (-3.22 + 2.07i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (-0.267 + 1.86i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (-0.804 - 1.76i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.112 + 0.780i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (4.90 - 5.66i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (-3.96 - 4.57i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (1.75 - 2.03i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-2.65 + 0.780i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (-8.29 - 5.33i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (5.52 - 3.55i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (7.72 + 2.26i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + 1.75T + 47T^{2} \)
53 \( 1 + (-0.516 + 3.59i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-0.999 - 6.95i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (1.86 - 0.548i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-4.08 + 8.93i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (2.88 - 6.32i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (8.63 + 9.96i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (-0.0541 - 0.376i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-2.75 - 1.77i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (4.66 + 1.36i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (7.65 - 4.91i)T + (40.2 - 88.2i)T^{2} \)
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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

−13.59533828799262227166548258374, −12.86649889208610436548562065043, −11.95875406718968042162951149081, −10.60114856126280382396405332628, −9.794446194878698942451476091695, −8.405109428856659841948652892900, −6.62115084655683187130605764926, −5.80201335226898935945923964467, −4.59453873606541407066044035972, −1.50865022472740446597481635932, 2.59150141901499813509101343603, 5.10168564920013819184625994019, 5.95976774845305504930063821820, 6.93470564960570300520293855155, 9.110201356814059010788575342907, 9.939354968364120202891013436115, 11.22955004809419568602540226454, 11.59456128883190743132595718929, 13.39236866483717195331757385979, 14.02420781845998700649123600962

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