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Properties

Label 2-92-23.6-c1-0-0
Degree $2$
Conductor $92$
Sign $0.973 - 0.229i$
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  + (1.69 + 0.496i)3-s + (−0.630 + 0.404i)5-s + (−0.0283 + 0.197i)7-s + (0.0904 + 0.0581i)9-s + (−1.64 − 3.59i)11-s + (0.588 + 4.09i)13-s + (−1.26 + 0.371i)15-s + (1.78 − 2.06i)17-s + (−2.52 − 2.91i)19-s + (−0.145 + 0.319i)21-s + (−4.78 − 0.377i)23-s + (−1.84 + 4.03i)25-s + (−3.33 − 3.85i)27-s + (−4.13 + 4.77i)29-s + (5.69 − 1.67i)31-s + ⋯
L(s)  = 1  + (0.976 + 0.286i)3-s + (−0.281 + 0.181i)5-s + (−0.0107 + 0.0745i)7-s + (0.0301 + 0.0193i)9-s + (−0.495 − 1.08i)11-s + (0.163 + 1.13i)13-s + (−0.327 + 0.0960i)15-s + (0.433 − 0.500i)17-s + (−0.580 − 0.669i)19-s + (−0.0318 + 0.0697i)21-s + (−0.996 − 0.0786i)23-s + (−0.368 + 0.807i)25-s + (−0.642 − 0.741i)27-s + (−0.767 + 0.886i)29-s + (1.02 − 0.300i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $0.973 - 0.229i$
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: \((2,\ 92,\ (\ :1/2),\ 0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17722 + 0.137135i\)
\(L(\frac12)\) \(\approx\) \(1.17722 + 0.137135i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.78 + 0.377i)T \)
good3 \( 1 + (-1.69 - 0.496i)T + (2.52 + 1.62i)T^{2} \)
5 \( 1 + (0.630 - 0.404i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (0.0283 - 0.197i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (1.64 + 3.59i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-0.588 - 4.09i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-1.78 + 2.06i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (2.52 + 2.91i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (4.13 - 4.77i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-5.69 + 1.67i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (-6.76 - 4.34i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (-5.51 + 3.54i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-6.37 - 1.87i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 - 1.22T + 47T^{2} \)
53 \( 1 + (-0.705 + 4.90i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-1.64 - 11.4i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (11.9 - 3.51i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (0.860 - 1.88i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-3.58 + 7.85i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-0.122 - 0.141i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (0.139 + 0.969i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-4.66 - 2.99i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (16.2 + 4.77i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (-11.6 + 7.48i)T + (40.2 - 88.2i)T^{2} \)
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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

−14.08372721445037261763386541553, −13.42333768608785995664454226343, −11.87193404681880207505009273787, −10.92735637464390711730693217457, −9.478097263869950362679538548848, −8.661457577175244552670671989079, −7.56118190246535884407323955826, −5.97185065794592753111109248202, −4.09905190872417244999117899713, −2.75816976085574177475349359227, 2.39888837798450508560481424104, 4.07707823344088101802071971990, 5.86243990299274449428959368105, 7.76344728662076054487694878471, 8.098889356520693972809352303905, 9.613930989261910579431528205584, 10.63583806520366143951668148963, 12.23191943429873749397530571865, 12.97906395525829172355604996941, 14.07217524778304042491602042473

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