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Properties

Label 2-2023-119.118-c0-0-3
Degree 22
Conductor 20232023
Sign 0.410+0.911i0.410 + 0.911i
Analytic cond. 1.009601.00960
Root an. cond. 1.004791.00479
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  − 1.53·2-s + 1.34·4-s + i·7-s − 0.532·8-s − 9-s − i·11-s − 1.53i·14-s − 0.532·16-s + 1.53·18-s + 1.53i·22-s − 1.87i·23-s − 25-s + 1.34i·28-s − 0.347i·29-s + 1.34·32-s + ⋯
L(s)  = 1  − 1.53·2-s + 1.34·4-s + i·7-s − 0.532·8-s − 9-s − i·11-s − 1.53i·14-s − 0.532·16-s + 1.53·18-s + 1.53i·22-s − 1.87i·23-s − 25-s + 1.34i·28-s − 0.347i·29-s + 1.34·32-s + ⋯

Functional equation

Λ(s)=(2023s/2 ΓC(s) L(s)=((0.410+0.911i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2023s/2 ΓC(s) L(s)=((0.410+0.911i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20232023    =    7⋅1727 \cdot 17^{2}
Sign: 0.410+0.911i0.410 + 0.911i
Analytic conductor: 1.009601.00960
Root analytic conductor: 1.004791.00479
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2023(2022,⋅)\chi_{2023} (2022, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2023, ( :0), 0.410+0.911i)(2,\ 2023,\ (\ :0),\ 0.410 + 0.911i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.38406410730.3840641073
L(12)L(\frac12) ≈\approx 0.38406410730.3840641073
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)
bad7 1−iT 1 - iT
17 1 1
good2 1+1.53T+T2 1 + 1.53T + T^{2}
3 1+T2 1 + T^{2}
5 1+T2 1 + T^{2}
11 1+iT−T2 1 + iT - T^{2}
13 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+1.87iT−T2 1 + 1.87iT - T^{2}
29 1+0.347iT−T2 1 + 0.347iT - T^{2}
31 1+T2 1 + T^{2}
37 1+1.53iT−T2 1 + 1.53iT - T^{2}
41 1+T2 1 + T^{2}
43 1−1.87T+T2 1 - 1.87T + T^{2}
47 1−T2 1 - T^{2}
53 1−1.87T+T2 1 - 1.87T + T^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1+T+T2 1 + T + T^{2}
71 1+0.347iT−T2 1 + 0.347iT - T^{2}
73 1+T2 1 + T^{2}
79 1−1.53iT−T2 1 - 1.53iT - T^{2}
83 1−T2 1 - T^{2}
89 1−T2 1 - T^{2}
97 1+T2 1 + 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

−9.012559155119754646807027707724, −8.532832333607967656260370230588, −8.068473483121951445617456962171, −7.07282892232719285744953670715, −6.03283883601741273295377449881, −5.63249123993757475067413102078, −4.21694942121274496795247682812, −2.80438526837903144177436316749, −2.17972705380269107249164440375, −0.50520965818687737848697960532, 1.18655545695114325122636657150, 2.26410997869816283577827879705, 3.53630148874755540983525967819, 4.59092515862668573778498521511, 5.69878736270048725278712788943, 6.70566399828813917297009893091, 7.57213833493246970661989920368, 7.76771310776585383755212617436, 8.847201325608735977525591797187, 9.418485561582072036302671788812

Graph of the ZZ-function along the critical line