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Properties

Label 2-92-92.15-c1-0-4
Degree 22
Conductor 9292
Sign 0.577−0.816i0.577 - 0.816i
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  + (0.0506 + 1.41i)2-s + (1.95 − 0.891i)3-s + (−1.99 + 0.143i)4-s + (−1.03 + 0.895i)5-s + (1.35 + 2.71i)6-s + (2.14 + 1.38i)7-s + (−0.303 − 2.81i)8-s + (1.04 − 1.21i)9-s + (−1.31 − 1.41i)10-s + (−0.745 − 5.18i)11-s + (−3.76 + 2.05i)12-s + (−3.68 + 2.37i)13-s + (−1.84 + 3.10i)14-s + (−1.21 + 2.66i)15-s + (3.95 − 0.571i)16-s + (−0.725 − 2.46i)17-s + ⋯
L(s)  = 1  + (0.0358 + 0.999i)2-s + (1.12 − 0.514i)3-s + (−0.997 + 0.0715i)4-s + (−0.461 + 0.400i)5-s + (0.554 + 1.10i)6-s + (0.811 + 0.521i)7-s + (−0.107 − 0.994i)8-s + (0.349 − 0.403i)9-s + (−0.416 − 0.447i)10-s + (−0.224 − 1.56i)11-s + (−1.08 + 0.593i)12-s + (−1.02 + 0.657i)13-s + (−0.492 + 0.829i)14-s + (−0.314 + 0.688i)15-s + (0.989 − 0.142i)16-s + (−0.175 − 0.598i)17-s + ⋯

Functional equation

Λ(s)=(92s/2 ΓC(s) L(s)=((0.577−0.816i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(92s/2 ΓC(s+1/2) L(s)=((0.577−0.816i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9292    =    22⋅232^{2} \cdot 23
Sign: 0.577−0.816i0.577 - 0.816i
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), 0.577−0.816i)(2,\ 92,\ (\ :1/2),\ 0.577 - 0.816i)

Particular Values

L(1)L(1) ≈\approx 1.05165+0.544593i1.05165 + 0.544593i
L(12)L(\frac12) ≈\approx 1.05165+0.544593i1.05165 + 0.544593i
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+(−0.0506−1.41i)T 1 + (-0.0506 - 1.41i)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

−14.23853046128078352346731319267, −13.75374961393653118608397078528, −12.40533598833556161628366404832, −11.06962585393885949498262365507, −9.256213579052269907285696975184, −8.340517423796097276391225870094, −7.72334660618335970756772908688, −6.40456715677746036514743919782, −4.72926229481682348132017256107, −2.88266816260284670210028246568, 2.27282752084962362412821926846, 4.00294090932685059375795407283, 4.82415942749462416984295940115, 7.71249699776801153223336948074, 8.460130170548704165902577841040, 9.759897317207274418110936947410, 10.40118004125477619741502054884, 11.91368242812691401116835330986, 12.71991526058025249032069050317, 13.95829897538232068935995516089

Graph of the ZZ-function along the critical line