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Properties

Label 2-1805-95.42-c0-0-0
Degree 22
Conductor 18051805
Sign −0.709+0.704i-0.709 + 0.704i
Analytic cond. 0.9008120.900812
Root an. cond. 0.9491110.949111
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.642 − 0.766i)4-s + (−0.766 + 0.642i)5-s + (−1.36 + 0.366i)7-s + (−0.342 − 0.939i)9-s + (−0.173 − 0.984i)16-s + (−0.597 − 1.28i)17-s + i·20-s + (−1.40 + 0.123i)23-s + (0.173 − 0.984i)25-s + (−0.597 + 1.28i)28-s + (0.811 − 1.15i)35-s + (−0.939 − 0.342i)36-s + (0.123 − 1.40i)43-s + (0.866 + 0.5i)45-s + (−1.28 − 0.597i)47-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)4-s + (−0.766 + 0.642i)5-s + (−1.36 + 0.366i)7-s + (−0.342 − 0.939i)9-s + (−0.173 − 0.984i)16-s + (−0.597 − 1.28i)17-s + i·20-s + (−1.40 + 0.123i)23-s + (0.173 − 0.984i)25-s + (−0.597 + 1.28i)28-s + (0.811 − 1.15i)35-s + (−0.939 − 0.342i)36-s + (0.123 − 1.40i)43-s + (0.866 + 0.5i)45-s + (−1.28 − 0.597i)47-s + ⋯

Functional equation

Λ(s)=(1805s/2 ΓC(s) L(s)=((−0.709+0.704i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1805s/2 ΓC(s) L(s)=((−0.709+0.704i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18051805    =    5⋅1925 \cdot 19^{2}
Sign: −0.709+0.704i-0.709 + 0.704i
Analytic conductor: 0.9008120.900812
Root analytic conductor: 0.9491110.949111
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1805(1182,⋅)\chi_{1805} (1182, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1805, ( :0), −0.709+0.704i)(2,\ 1805,\ (\ :0),\ -0.709 + 0.704i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.50308463960.5030846396
L(12)L(\frac12) ≈\approx 0.50308463960.5030846396
L(1)L(1) 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+(0.766−0.642i)T 1 + (0.766 - 0.642i)T
19 1 1
good2 1+(−0.642+0.766i)T2 1 + (-0.642 + 0.766i)T^{2}
3 1+(0.342+0.939i)T2 1 + (0.342 + 0.939i)T^{2}
7 1+(1.36−0.366i)T+(0.866−0.5i)T2 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2}
11 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
13 1+(−0.342+0.939i)T2 1 + (-0.342 + 0.939i)T^{2}
17 1+(0.597+1.28i)T+(−0.642+0.766i)T2 1 + (0.597 + 1.28i)T + (-0.642 + 0.766i)T^{2}
23 1+(1.40−0.123i)T+(0.984−0.173i)T2 1 + (1.40 - 0.123i)T + (0.984 - 0.173i)T^{2}
29 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
31 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
37 1+iT2 1 + iT^{2}
41 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
43 1+(−0.123+1.40i)T+(−0.984−0.173i)T2 1 + (-0.123 + 1.40i)T + (-0.984 - 0.173i)T^{2}
47 1+(1.28+0.597i)T+(0.642+0.766i)T2 1 + (1.28 + 0.597i)T + (0.642 + 0.766i)T^{2}
53 1+(0.984−0.173i)T2 1 + (0.984 - 0.173i)T^{2}
59 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
61 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
67 1+(0.642+0.766i)T2 1 + (0.642 + 0.766i)T^{2}
71 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
73 1+(−1.15−0.811i)T+(0.342+0.939i)T2 1 + (-1.15 - 0.811i)T + (0.342 + 0.939i)T^{2}
79 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
83 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
89 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
97 1+(−0.642+0.766i)T2 1 + (-0.642 + 0.766i)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

−9.440286355510871107092902891411, −8.433407425495988858717409555262, −7.28761811285294807981549047801, −6.67364822568785863254098024173, −6.22974114870588606817100308735, −5.30416186258105485323878057504, −3.93690940540000200548673935909, −3.12371103530665380156743823547, −2.33865530084199106035105353313, −0.34005321573832572035404083816, 1.91634048152943944396380247678, 3.09753847322977760940785880775, 3.84346253308110424663653513014, 4.62533877399846836748248385594, 6.00146535301962561152848881374, 6.56125734627026317341798071391, 7.60122289464399661086440974086, 8.071166358392073670213800357888, 8.754106516778917863821193659940, 9.767548246742453126574164379643

Graph of the ZZ-function along the critical line