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Properties

Label 2-1805-5.4-c1-0-148
Degree 22
Conductor 18051805
Sign 0.525−0.851i0.525 - 0.851i
Analytic cond. 14.412914.4129
Root an. cond. 3.796443.79644
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  − 2.37i·2-s − 2.28i·3-s − 3.62·4-s + (1.17 − 1.90i)5-s − 5.41·6-s − 1.63i·7-s + 3.84i·8-s − 2.22·9-s + (−4.51 − 2.78i)10-s + 2.72·11-s + 8.27i·12-s − 6.19i·13-s − 3.88·14-s + (−4.34 − 2.68i)15-s + 1.87·16-s + 3.12i·17-s + ⋯
L(s)  = 1  − 1.67i·2-s − 1.31i·3-s − 1.81·4-s + (0.525 − 0.851i)5-s − 2.21·6-s − 0.618i·7-s + 1.35i·8-s − 0.740·9-s + (−1.42 − 0.880i)10-s + 0.822·11-s + 2.38i·12-s − 1.71i·13-s − 1.03·14-s + (−1.12 − 0.692i)15-s + 0.468·16-s + 0.757i·17-s + ⋯

Functional equation

Λ(s)=(1805s/2 ΓC(s) L(s)=((0.525−0.851i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1805s/2 ΓC(s+1/2) L(s)=((0.525−0.851i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18051805    =    5⋅1925 \cdot 19^{2}
Sign: 0.525−0.851i0.525 - 0.851i
Analytic conductor: 14.412914.4129
Root analytic conductor: 3.796443.79644
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1805(1084,⋅)\chi_{1805} (1084, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1805, ( :1/2), 0.525−0.851i)(2,\ 1805,\ (\ :1/2),\ 0.525 - 0.851i)

Particular Values

L(1)L(1) ≈\approx 1.6028700421.602870042
L(12)L(\frac12) ≈\approx 1.6028700421.602870042
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)
bad5 1+(−1.17+1.90i)T 1 + (-1.17 + 1.90i)T
19 1 1
good2 1+2.37iT−2T2 1 + 2.37iT - 2T^{2}
3 1+2.28iT−3T2 1 + 2.28iT - 3T^{2}
7 1+1.63iT−7T2 1 + 1.63iT - 7T^{2}
11 1−2.72T+11T2 1 - 2.72T + 11T^{2}
13 1+6.19iT−13T2 1 + 6.19iT - 13T^{2}
17 1−3.12iT−17T2 1 - 3.12iT - 17T^{2}
23 1−7.29iT−23T2 1 - 7.29iT - 23T^{2}
29 1−2.22T+29T2 1 - 2.22T + 29T^{2}
31 1+4.42T+31T2 1 + 4.42T + 31T^{2}
37 1+2.04iT−37T2 1 + 2.04iT - 37T^{2}
41 1+3.92T+41T2 1 + 3.92T + 41T^{2}
43 1−0.472iT−43T2 1 - 0.472iT - 43T^{2}
47 1+2.30iT−47T2 1 + 2.30iT - 47T^{2}
53 1−6.36iT−53T2 1 - 6.36iT - 53T^{2}
59 1−12.2T+59T2 1 - 12.2T + 59T^{2}
61 1+4.79T+61T2 1 + 4.79T + 61T^{2}
67 1+0.670iT−67T2 1 + 0.670iT - 67T^{2}
71 1+2.63T+71T2 1 + 2.63T + 71T^{2}
73 1+6.51iT−73T2 1 + 6.51iT - 73T^{2}
79 1−4.12T+79T2 1 - 4.12T + 79T^{2}
83 1−6.42iT−83T2 1 - 6.42iT - 83T^{2}
89 1−17.3T+89T2 1 - 17.3T + 89T^{2}
97 1+0.129iT−97T2 1 + 0.129iT - 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

−8.746469918845058592263075016968, −8.008101148675212778355503552177, −7.18765108192087934825902793801, −6.06412609235843817168199534034, −5.26868508633411251648756836535, −4.09707414226647675857661574666, −3.28168885329386409260578309301, −2.06233069342648417286413423828, −1.32743162507858959272267462192, −0.64801841322893468191182777336, 2.25914036541257619747687183358, 3.64758726477908918185970993341, 4.49117014399852409895384651366, 5.12470817319213155883932476797, 6.09403297037601781002353507524, 6.68238902739053807602292938448, 7.23355620976571378626441684510, 8.587333693042207957735279163624, 9.062682781129170944900368318620, 9.565313944995108044807449219000

Graph of the ZZ-function along the critical line