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Properties

Label 2-363-11.3-c1-0-9
Degree 22
Conductor 363363
Sign 0.751+0.659i0.751 + 0.659i
Analytic cond. 2.898562.89856
Root an. cond. 1.702511.70251
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  + (−2.11 + 1.53i)2-s + (−0.309 − 0.951i)3-s + (1.5 − 4.61i)4-s + (0.5 + 0.363i)5-s + (2.11 + 1.53i)6-s + (0.309 − 0.951i)7-s + (2.30 + 7.10i)8-s + (−0.809 + 0.587i)9-s − 1.61·10-s − 4.85·12-s + (0.190 − 0.138i)13-s + (0.809 + 2.48i)14-s + (0.190 − 0.587i)15-s + (−7.97 − 5.79i)16-s + (−0.927 − 0.673i)17-s + (0.809 − 2.48i)18-s + ⋯
L(s)  = 1  + (−1.49 + 1.08i)2-s + (−0.178 − 0.549i)3-s + (0.750 − 2.30i)4-s + (0.223 + 0.162i)5-s + (0.864 + 0.628i)6-s + (0.116 − 0.359i)7-s + (0.816 + 2.51i)8-s + (−0.269 + 0.195i)9-s − 0.511·10-s − 1.40·12-s + (0.0529 − 0.0384i)13-s + (0.216 + 0.665i)14-s + (0.0493 − 0.151i)15-s + (−1.99 − 1.44i)16-s + (−0.224 − 0.163i)17-s + (0.190 − 0.586i)18-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=((0.751+0.659i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=((0.751+0.659i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: 0.751+0.659i0.751 + 0.659i
Analytic conductor: 2.898562.89856
Root analytic conductor: 1.702511.70251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ363(124,⋅)\chi_{363} (124, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 363, ( :1/2), 0.751+0.659i)(2,\ 363,\ (\ :1/2),\ 0.751 + 0.659i)

Particular Values

L(1)L(1) ≈\approx 0.489157−0.184075i0.489157 - 0.184075i
L(12)L(\frac12) ≈\approx 0.489157−0.184075i0.489157 - 0.184075i
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)
bad3 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
11 1 1
good2 1+(2.11−1.53i)T+(0.618−1.90i)T2 1 + (2.11 - 1.53i)T + (0.618 - 1.90i)T^{2}
5 1+(−0.5−0.363i)T+(1.54+4.75i)T2 1 + (-0.5 - 0.363i)T + (1.54 + 4.75i)T^{2}
7 1+(−0.309+0.951i)T+(−5.66−4.11i)T2 1 + (-0.309 + 0.951i)T + (-5.66 - 4.11i)T^{2}
13 1+(−0.190+0.138i)T+(4.01−12.3i)T2 1 + (-0.190 + 0.138i)T + (4.01 - 12.3i)T^{2}
17 1+(0.927+0.673i)T+(5.25+16.1i)T2 1 + (0.927 + 0.673i)T + (5.25 + 16.1i)T^{2}
19 1+(1.80+5.56i)T+(−15.3+11.1i)T2 1 + (1.80 + 5.56i)T + (-15.3 + 11.1i)T^{2}
23 1−0.236T+23T2 1 - 0.236T + 23T^{2}
29 1+(−1.85+5.70i)T+(−23.4−17.0i)T2 1 + (-1.85 + 5.70i)T + (-23.4 - 17.0i)T^{2}
31 1+(−4.92+3.57i)T+(9.57−29.4i)T2 1 + (-4.92 + 3.57i)T + (9.57 - 29.4i)T^{2}
37 1+(1.92−5.93i)T+(−29.9−21.7i)T2 1 + (1.92 - 5.93i)T + (-29.9 - 21.7i)T^{2}
41 1+(0.0729+0.224i)T+(−33.1+24.0i)T2 1 + (0.0729 + 0.224i)T + (-33.1 + 24.0i)T^{2}
43 1−6.70T+43T2 1 - 6.70T + 43T^{2}
47 1+(3.11+9.59i)T+(−38.0+27.6i)T2 1 + (3.11 + 9.59i)T + (-38.0 + 27.6i)T^{2}
53 1+(−0.309+0.224i)T+(16.3−50.4i)T2 1 + (-0.309 + 0.224i)T + (16.3 - 50.4i)T^{2}
59 1+(−2.28+7.02i)T+(−47.7−34.6i)T2 1 + (-2.28 + 7.02i)T + (-47.7 - 34.6i)T^{2}
61 1+(9.35+6.79i)T+(18.8+58.0i)T2 1 + (9.35 + 6.79i)T + (18.8 + 58.0i)T^{2}
67 1−1.85T+67T2 1 - 1.85T + 67T^{2}
71 1+(8.35+6.06i)T+(21.9+67.5i)T2 1 + (8.35 + 6.06i)T + (21.9 + 67.5i)T^{2}
73 1+(−1.76+5.42i)T+(−59.0−42.9i)T2 1 + (-1.76 + 5.42i)T + (-59.0 - 42.9i)T^{2}
79 1+(−8.89+6.46i)T+(24.4−75.1i)T2 1 + (-8.89 + 6.46i)T + (24.4 - 75.1i)T^{2}
83 1+(−1.19−0.865i)T+(25.6+78.9i)T2 1 + (-1.19 - 0.865i)T + (25.6 + 78.9i)T^{2}
89 1+8.23T+89T2 1 + 8.23T + 89T^{2}
97 1+(6.35−4.61i)T+(29.9−92.2i)T2 1 + (6.35 - 4.61i)T + (29.9 - 92.2i)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

−10.96261400251855302843839548763, −10.20024449882396976510623672915, −9.297226099426997506362517302827, −8.364320332696430081101218688701, −7.63068183853984660780592469763, −6.67875773645983708311433837772, −6.12797290465361444543822878569, −4.78225054740279230987821264256, −2.26750602750295708225563223956, −0.61893711627629929726786056548, 1.51156156693103737334209707350, 2.90379195700167494169822798263, 4.12845160631002208742763406072, 5.73836767489756991819298644282, 7.18073510245233359133889444389, 8.314039392246169966491289403067, 8.952095025784348930146286375358, 9.772333479153175490631149122071, 10.54404839818605914599995620682, 11.17638069045067913678476796520

Graph of the ZZ-function along the critical line