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Properties

Label 2-819-13.12-c1-0-33
Degree 22
Conductor 819819
Sign 0.429−0.903i0.429 - 0.903i
Analytic cond. 6.539746.53974
Root an. cond. 2.557292.55729
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  − 2.54i·2-s − 4.49·4-s − 3.49i·5-s + i·7-s + 6.35i·8-s − 8.90·10-s + 0.708i·11-s + (−3.25 − 1.54i)13-s + 2.54·14-s + 7.20·16-s − 7.09·17-s − 0.311i·19-s + 15.6i·20-s + 1.80·22-s + 7.88·23-s + ⋯
L(s)  = 1  − 1.80i·2-s − 2.24·4-s − 1.56i·5-s + 0.377i·7-s + 2.24i·8-s − 2.81·10-s + 0.213i·11-s + (−0.903 − 0.429i)13-s + 0.681·14-s + 1.80·16-s − 1.72·17-s − 0.0714i·19-s + 3.50i·20-s + 0.384·22-s + 1.64·23-s + ⋯

Functional equation

Λ(s)=(819s/2 ΓC(s) L(s)=((0.429−0.903i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(819s/2 ΓC(s+1/2) L(s)=((0.429−0.903i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 819819    =    32⋅7⋅133^{2} \cdot 7 \cdot 13
Sign: 0.429−0.903i0.429 - 0.903i
Analytic conductor: 6.539746.53974
Root analytic conductor: 2.557292.55729
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ819(64,⋅)\chi_{819} (64, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 819, ( :1/2), 0.429−0.903i)(2,\ 819,\ (\ :1/2),\ 0.429 - 0.903i)

Particular Values

L(1)L(1) ≈\approx 0.455557+0.287835i0.455557 + 0.287835i
L(12)L(\frac12) ≈\approx 0.455557+0.287835i0.455557 + 0.287835i
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 1
7 1−iT 1 - iT
13 1+(3.25+1.54i)T 1 + (3.25 + 1.54i)T
good2 1+2.54iT−2T2 1 + 2.54iT - 2T^{2}
5 1+3.49iT−5T2 1 + 3.49iT - 5T^{2}
11 1−0.708iT−11T2 1 - 0.708iT - 11T^{2}
17 1+7.09T+17T2 1 + 7.09T + 17T^{2}
19 1+0.311iT−19T2 1 + 0.311iT - 19T^{2}
23 1−7.88T+23T2 1 - 7.88T + 23T^{2}
29 1+5.29T+29T2 1 + 5.29T + 29T^{2}
31 1−7.29iT−31T2 1 - 7.29iT - 31T^{2}
37 1−1.41iT−37T2 1 - 1.41iT - 37T^{2}
41 1+11.8iT−41T2 1 + 11.8iT - 41T^{2}
43 1+3.29T+43T2 1 + 3.29T + 43T^{2}
47 1+6.11iT−47T2 1 + 6.11iT - 47T^{2}
53 1−3.72T+53T2 1 - 3.72T + 53T^{2}
59 1−2.19iT−59T2 1 - 2.19iT - 59T^{2}
61 1−2.51T+61T2 1 - 2.51T + 61T^{2}
67 1+9.17iT−67T2 1 + 9.17iT - 67T^{2}
71 1+0.708iT−71T2 1 + 0.708iT - 71T^{2}
73 1−5.21iT−73T2 1 - 5.21iT - 73T^{2}
79 1+2.78T+79T2 1 + 2.78T + 79T^{2}
83 1−6.11iT−83T2 1 - 6.11iT - 83T^{2}
89 1+11.1iT−89T2 1 + 11.1iT - 89T^{2}
97 1+7.79iT−97T2 1 + 7.79iT - 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

−9.534335490034554833608576896603, −8.902809328496816531641531379651, −8.555664576226750169366970345842, −7.07613454291346259561961764125, −5.24194871744925229990154578699, −4.90873530851084913430746204464, −3.91373746713489989598581501150, −2.60736575609690931551559552239, −1.61312846008968396303495662732, −0.26434130634099849361488727571, 2.62566912790042944580214907855, 4.00059757513308880187187222175, 4.90762006489605681850315338799, 6.08744969766922910747544874051, 6.79739606727227822574197868169, 7.18950193366055775808636631738, 8.002808330528769329267706802437, 9.120424566650464029709399687286, 9.745606533213831873449132776533, 10.89416218357201781469201790348

Graph of the ZZ-function along the critical line