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Properties

Label 2-3762-209.208-c1-0-83
Degree 22
Conductor 37623762
Sign 0.902+0.431i0.902 + 0.431i
Analytic cond. 30.039730.0397
Root an. cond. 5.480855.48085
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-s + 4-s + 3.98·5-s + 0.391i·7-s + 8-s + 3.98·10-s + (−2.34 − 2.34i)11-s + 0.870·13-s + 0.391i·14-s + 16-s − 5.35i·17-s + (−4.11 + 1.45i)19-s + 3.98·20-s + (−2.34 − 2.34i)22-s − 2.08·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.78·5-s + 0.148i·7-s + 0.353·8-s + 1.26·10-s + (−0.706 − 0.707i)11-s + 0.241·13-s + 0.104i·14-s + 0.250·16-s − 1.29i·17-s + (−0.942 + 0.332i)19-s + 0.890·20-s + (−0.499 − 0.500i)22-s − 0.435·23-s + ⋯

Functional equation

Λ(s)=(3762s/2 ΓC(s) L(s)=((0.902+0.431i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3762s/2 ΓC(s+1/2) L(s)=((0.902+0.431i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37623762    =    2⋅32⋅11⋅192 \cdot 3^{2} \cdot 11 \cdot 19
Sign: 0.902+0.431i0.902 + 0.431i
Analytic conductor: 30.039730.0397
Root analytic conductor: 5.480855.48085
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3762(2089,⋅)\chi_{3762} (2089, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3762, ( :1/2), 0.902+0.431i)(2,\ 3762,\ (\ :1/2),\ 0.902 + 0.431i)

Particular Values

L(1)L(1) ≈\approx 4.2218865284.221886528
L(12)L(\frac12) ≈\approx 4.2218865284.221886528
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−T 1 - T
3 1 1
11 1+(2.34+2.34i)T 1 + (2.34 + 2.34i)T
19 1+(4.11−1.45i)T 1 + (4.11 - 1.45i)T
good5 1−3.98T+5T2 1 - 3.98T + 5T^{2}
7 1−0.391iT−7T2 1 - 0.391iT - 7T^{2}
13 1−0.870T+13T2 1 - 0.870T + 13T^{2}
17 1+5.35iT−17T2 1 + 5.35iT - 17T^{2}
23 1+2.08T+23T2 1 + 2.08T + 23T^{2}
29 1−8.28T+29T2 1 - 8.28T + 29T^{2}
31 1−5.16iT−31T2 1 - 5.16iT - 31T^{2}
37 1+7.83iT−37T2 1 + 7.83iT - 37T^{2}
41 1−3.44T+41T2 1 - 3.44T + 41T^{2}
43 1+11.5iT−43T2 1 + 11.5iT - 43T^{2}
47 1−12.6T+47T2 1 - 12.6T + 47T^{2}
53 1+1.51iT−53T2 1 + 1.51iT - 53T^{2}
59 1−11.2iT−59T2 1 - 11.2iT - 59T^{2}
61 1+0.200iT−61T2 1 + 0.200iT - 61T^{2}
67 1+10.3iT−67T2 1 + 10.3iT - 67T^{2}
71 1−11.8iT−71T2 1 - 11.8iT - 71T^{2}
73 1−4.05iT−73T2 1 - 4.05iT - 73T^{2}
79 1+1.63T+79T2 1 + 1.63T + 79T^{2}
83 1+13.8iT−83T2 1 + 13.8iT - 83T^{2}
89 1−16.3iT−89T2 1 - 16.3iT - 89T^{2}
97 1−3.49iT−97T2 1 - 3.49iT - 97T^{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

−8.748372428809376608553196281086, −7.52179646826858107519626421709, −6.73931239460548748811003289830, −6.00226626383631920555276870832, −5.53172189756125725685552399407, −4.92069329107167799446601777192, −3.83265569395249578921401568858, −2.56146279338854123807723425064, −2.38810527591405728927607158963, −0.999449108982013741474382311164, 1.34735217620769386657166357511, 2.23684433528930805076978449124, 2.80090699225777594259582167303, 4.17031182477969605919234704234, 4.78028843035538509701699832277, 5.68735335008663519297338347640, 6.22236527935716039081426813062, 6.70035024860627580125654607706, 7.79604987114322653974324663138, 8.542158956703509139640255907035

Graph of the ZZ-function along the critical line