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Properties

Label 2-92-92.15-c1-0-0
Degree 22
Conductor 9292
Sign −0.971+0.238i-0.971 + 0.238i
Analytic cond. 0.7346230.734623
Root an. cond. 0.8571010.857101
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.446 + 1.34i)2-s + (−1.95 + 0.891i)3-s + (−1.60 − 1.19i)4-s + (−1.03 + 0.895i)5-s + (−0.323 − 3.01i)6-s + (−2.14 − 1.38i)7-s + (2.32 − 1.61i)8-s + (1.04 − 1.21i)9-s + (−0.739 − 1.78i)10-s + (0.745 + 5.18i)11-s + (4.19 + 0.913i)12-s + (−3.68 + 2.37i)13-s + (2.81 − 2.26i)14-s + (1.21 − 2.66i)15-s + (1.12 + 3.83i)16-s + (−0.725 − 2.46i)17-s + ⋯
L(s)  = 1  + (−0.315 + 0.948i)2-s + (−1.12 + 0.514i)3-s + (−0.800 − 0.599i)4-s + (−0.461 + 0.400i)5-s + (−0.132 − 1.23i)6-s + (−0.811 − 0.521i)7-s + (0.821 − 0.570i)8-s + (0.349 − 0.403i)9-s + (−0.233 − 0.564i)10-s + (0.224 + 1.56i)11-s + (1.21 + 0.263i)12-s + (−1.02 + 0.657i)13-s + (0.751 − 0.605i)14-s + (0.314 − 0.688i)15-s + (0.281 + 0.959i)16-s + (−0.175 − 0.598i)17-s + ⋯

Functional equation

Λ(s)=(92s/2 ΓC(s) L(s)=((−0.971+0.238i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(92s/2 ΓC(s+1/2) L(s)=((−0.971+0.238i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9292    =    22⋅232^{2} \cdot 23
Sign: −0.971+0.238i-0.971 + 0.238i
Analytic conductor: 0.7346230.734623
Root analytic conductor: 0.8571010.857101
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ92(15,⋅)\chi_{92} (15, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 92, ( :1/2), −0.971+0.238i)(2,\ 92,\ (\ :1/2),\ -0.971 + 0.238i)

Particular Values

L(1)L(1) ≈\approx 0.0369143−0.304636i0.0369143 - 0.304636i
L(12)L(\frac12) ≈\approx 0.0369143−0.304636i0.0369143 - 0.304636i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.446−1.34i)T 1 + (0.446 - 1.34i)T
23 1+(−1.53−4.54i)T 1 + (-1.53 - 4.54i)T
good3 1+(1.95−0.891i)T+(1.96−2.26i)T2 1 + (1.95 - 0.891i)T + (1.96 - 2.26i)T^{2}
5 1+(1.03−0.895i)T+(0.711−4.94i)T2 1 + (1.03 - 0.895i)T + (0.711 - 4.94i)T^{2}
7 1+(2.14+1.38i)T+(2.90+6.36i)T2 1 + (2.14 + 1.38i)T + (2.90 + 6.36i)T^{2}
11 1+(−0.745−5.18i)T+(−10.5+3.09i)T2 1 + (-0.745 - 5.18i)T + (-10.5 + 3.09i)T^{2}
13 1+(3.68−2.37i)T+(5.40−11.8i)T2 1 + (3.68 - 2.37i)T + (5.40 - 11.8i)T^{2}
17 1+(0.725+2.46i)T+(−14.3+9.19i)T2 1 + (0.725 + 2.46i)T + (-14.3 + 9.19i)T^{2}
19 1+(−2.83−0.832i)T+(15.9+10.2i)T2 1 + (-2.83 - 0.832i)T + (15.9 + 10.2i)T^{2}
29 1+(−4.83+1.42i)T+(24.3−15.6i)T2 1 + (-4.83 + 1.42i)T + (24.3 - 15.6i)T^{2}
31 1+(5.16+2.35i)T+(20.3+23.4i)T2 1 + (5.16 + 2.35i)T + (20.3 + 23.4i)T^{2}
37 1+(0.714+0.618i)T+(5.26+36.6i)T2 1 + (0.714 + 0.618i)T + (5.26 + 36.6i)T^{2}
41 1+(−3.41−3.94i)T+(−5.83+40.5i)T2 1 + (-3.41 - 3.94i)T + (-5.83 + 40.5i)T^{2}
43 1+(2.17+4.75i)T+(−28.1+32.4i)T2 1 + (2.17 + 4.75i)T + (-28.1 + 32.4i)T^{2}
47 1+6.04iT−47T2 1 + 6.04iT - 47T^{2}
53 1+(6.78−10.5i)T+(−22.0−48.2i)T2 1 + (6.78 - 10.5i)T + (-22.0 - 48.2i)T^{2}
59 1+(−2.90−4.51i)T+(−24.5+53.6i)T2 1 + (-2.90 - 4.51i)T + (-24.5 + 53.6i)T^{2}
61 1+(1.40+0.643i)T+(39.9+46.1i)T2 1 + (1.40 + 0.643i)T + (39.9 + 46.1i)T^{2}
67 1+(2.03−14.1i)T+(−64.2−18.8i)T2 1 + (2.03 - 14.1i)T + (-64.2 - 18.8i)T^{2}
71 1+(−6.01−0.864i)T+(68.1+20.0i)T2 1 + (-6.01 - 0.864i)T + (68.1 + 20.0i)T^{2}
73 1+(−13.9−4.10i)T+(61.4+39.4i)T2 1 + (-13.9 - 4.10i)T + (61.4 + 39.4i)T^{2}
79 1+(1.15−0.739i)T+(32.8−71.8i)T2 1 + (1.15 - 0.739i)T + (32.8 - 71.8i)T^{2}
83 1+(6.21−7.17i)T+(−11.8−82.1i)T2 1 + (6.21 - 7.17i)T + (-11.8 - 82.1i)T^{2}
89 1+(6.34−2.89i)T+(58.2−67.2i)T2 1 + (6.34 - 2.89i)T + (58.2 - 67.2i)T^{2}
97 1+(−3.24+2.81i)T+(13.8−96.0i)T2 1 + (-3.24 + 2.81i)T + (13.8 - 96.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−15.00382297637725352668586775813, −13.82219633166900034155047133663, −12.43205007526003862208412533872, −11.32643548498073658527904758147, −10.00025427845120007345210191382, −9.530710209496395301889516810847, −7.38154138160215805815809586679, −6.85603782178984402766561245009, −5.32372606440200284184067988817, −4.23001873226722984842228268678, 0.46773010779926880147682104483, 3.16481878844097437489565146392, 5.09372875039499603534421125842, 6.39297946906820040788191254069, 8.084712540872461776231450141165, 9.188462661357770457656466855396, 10.57700772153545217232424840275, 11.45878841424288552045064960830, 12.43796440477046271988823503998, 12.76944856435903707617310525259

Graph of the ZZ-function along the critical line