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Properties

Label 2-368-92.15-c1-0-0
Degree 22
Conductor 368368
Sign −0.836−0.547i-0.836 - 0.547i
Analytic cond. 2.938492.93849
Root an. cond. 1.714201.71420
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  + (0.106 − 0.0488i)3-s + (−1.32 + 1.14i)5-s + (−2.88 − 1.85i)7-s + (−1.95 + 2.25i)9-s + (0.293 + 2.03i)11-s + (−3.33 + 2.14i)13-s + (−0.0852 + 0.186i)15-s + (0.842 + 2.87i)17-s + (−2.26 − 0.664i)19-s + (−0.399 − 0.0573i)21-s + (0.429 − 4.77i)23-s + (−0.277 + 1.92i)25-s + (−0.198 + 0.674i)27-s + (1.50 − 0.442i)29-s + (−7.01 − 3.20i)31-s + ⋯
L(s)  = 1  + (0.0617 − 0.0281i)3-s + (−0.590 + 0.511i)5-s + (−1.09 − 0.701i)7-s + (−0.651 + 0.752i)9-s + (0.0884 + 0.614i)11-s + (−0.924 + 0.594i)13-s + (−0.0220 + 0.0482i)15-s + (0.204 + 0.696i)17-s + (−0.518 − 0.152i)19-s + (−0.0870 − 0.0125i)21-s + (0.0894 − 0.995i)23-s + (−0.0554 + 0.385i)25-s + (−0.0381 + 0.129i)27-s + (0.280 − 0.0822i)29-s + (−1.26 − 0.575i)31-s + ⋯

Functional equation

Λ(s)=(368s/2 ΓC(s) L(s)=((−0.836−0.547i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(368s/2 ΓC(s+1/2) L(s)=((−0.836−0.547i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 368368    =    24⋅232^{4} \cdot 23
Sign: −0.836−0.547i-0.836 - 0.547i
Analytic conductor: 2.938492.93849
Root analytic conductor: 1.714201.71420
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ368(15,⋅)\chi_{368} (15, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 368, ( :1/2), −0.836−0.547i)(2,\ 368,\ (\ :1/2),\ -0.836 - 0.547i)

Particular Values

L(1)L(1) ≈\approx 0.120163+0.403195i0.120163 + 0.403195i
L(12)L(\frac12) ≈\approx 0.120163+0.403195i0.120163 + 0.403195i
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 1
23 1+(−0.429+4.77i)T 1 + (-0.429 + 4.77i)T
good3 1+(−0.106+0.0488i)T+(1.96−2.26i)T2 1 + (-0.106 + 0.0488i)T + (1.96 - 2.26i)T^{2}
5 1+(1.32−1.14i)T+(0.711−4.94i)T2 1 + (1.32 - 1.14i)T + (0.711 - 4.94i)T^{2}
7 1+(2.88+1.85i)T+(2.90+6.36i)T2 1 + (2.88 + 1.85i)T + (2.90 + 6.36i)T^{2}
11 1+(−0.293−2.03i)T+(−10.5+3.09i)T2 1 + (-0.293 - 2.03i)T + (-10.5 + 3.09i)T^{2}
13 1+(3.33−2.14i)T+(5.40−11.8i)T2 1 + (3.33 - 2.14i)T + (5.40 - 11.8i)T^{2}
17 1+(−0.842−2.87i)T+(−14.3+9.19i)T2 1 + (-0.842 - 2.87i)T + (-14.3 + 9.19i)T^{2}
19 1+(2.26+0.664i)T+(15.9+10.2i)T2 1 + (2.26 + 0.664i)T + (15.9 + 10.2i)T^{2}
29 1+(−1.50+0.442i)T+(24.3−15.6i)T2 1 + (-1.50 + 0.442i)T + (24.3 - 15.6i)T^{2}
31 1+(7.01+3.20i)T+(20.3+23.4i)T2 1 + (7.01 + 3.20i)T + (20.3 + 23.4i)T^{2}
37 1+(−6.47−5.60i)T+(5.26+36.6i)T2 1 + (-6.47 - 5.60i)T + (5.26 + 36.6i)T^{2}
41 1+(7.67+8.85i)T+(−5.83+40.5i)T2 1 + (7.67 + 8.85i)T + (-5.83 + 40.5i)T^{2}
43 1+(−4.27−9.36i)T+(−28.1+32.4i)T2 1 + (-4.27 - 9.36i)T + (-28.1 + 32.4i)T^{2}
47 1−3.80iT−47T2 1 - 3.80iT - 47T^{2}
53 1+(−2.11+3.29i)T+(−22.0−48.2i)T2 1 + (-2.11 + 3.29i)T + (-22.0 - 48.2i)T^{2}
59 1+(−2.94−4.58i)T+(−24.5+53.6i)T2 1 + (-2.94 - 4.58i)T + (-24.5 + 53.6i)T^{2}
61 1+(1.75+0.800i)T+(39.9+46.1i)T2 1 + (1.75 + 0.800i)T + (39.9 + 46.1i)T^{2}
67 1+(−0.399+2.78i)T+(−64.2−18.8i)T2 1 + (-0.399 + 2.78i)T + (-64.2 - 18.8i)T^{2}
71 1+(8.82+1.26i)T+(68.1+20.0i)T2 1 + (8.82 + 1.26i)T + (68.1 + 20.0i)T^{2}
73 1+(−7.48−2.19i)T+(61.4+39.4i)T2 1 + (-7.48 - 2.19i)T + (61.4 + 39.4i)T^{2}
79 1+(2.97−1.90i)T+(32.8−71.8i)T2 1 + (2.97 - 1.90i)T + (32.8 - 71.8i)T^{2}
83 1+(−6.57+7.59i)T+(−11.8−82.1i)T2 1 + (-6.57 + 7.59i)T + (-11.8 - 82.1i)T^{2}
89 1+(−7.44+3.39i)T+(58.2−67.2i)T2 1 + (-7.44 + 3.39i)T + (58.2 - 67.2i)T^{2}
97 1+(0.136−0.118i)T+(13.8−96.0i)T2 1 + (0.136 - 0.118i)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

−11.72388758471859677258628820724, −10.78013206039265224694605890167, −10.08406942837117556484938091021, −9.089332629580868000150140945765, −7.85390680603202947228696563159, −7.11295492916031999369531407643, −6.23405095782319707875095487384, −4.72596212180798991574851652207, −3.65078922697836931971561248826, −2.40201691734168770729577878119, 0.26402548698072396689646828368, 2.78416931008114353994901550517, 3.70666210281357500563600906385, 5.25622921636547692984666774918, 6.09855142238405933160308667504, 7.26099186539191053481157273208, 8.413552046677421303611938957551, 9.159220566908339866795781687365, 9.901438211507261837722297160227, 11.21010799842565776452775268565

Graph of the ZZ-function along the critical line