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Properties

Label 2-2023-7.6-c0-0-0
Degree 22
Conductor 20232023
Sign −i-i
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  − 0.618·2-s − 0.618i·3-s − 0.618·4-s + 1.61i·5-s + 0.381i·6-s + i·7-s + 8-s + 0.618·9-s − 1.00i·10-s + 0.381i·12-s − 0.618i·14-s + 1.00·15-s − 0.381·18-s − 0.999i·20-s + 0.618·21-s + ⋯
L(s)  = 1  − 0.618·2-s − 0.618i·3-s − 0.618·4-s + 1.61i·5-s + 0.381i·6-s + i·7-s + 8-s + 0.618·9-s − 1.00i·10-s + 0.381i·12-s − 0.618i·14-s + 1.00·15-s − 0.381·18-s − 0.999i·20-s + 0.618·21-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 20232023    =    7⋅1727 \cdot 17^{2}
Sign: −i-i
Analytic conductor: 1.009601.00960
Root analytic conductor: 1.004791.00479
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2023(1735,⋅)\chi_{2023} (1735, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2023, ( :0), −i)(2,\ 2023,\ (\ :0),\ -i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.68507365540.6850736554
L(12)L(\frac12) ≈\approx 0.68507365540.6850736554
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+0.618T+T2 1 + 0.618T + T^{2}
3 1+0.618iT−T2 1 + 0.618iT - T^{2}
5 1−1.61iT−T2 1 - 1.61iT - T^{2}
11 1+T2 1 + T^{2}
13 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+T2 1 + T^{2}
29 1+T2 1 + T^{2}
31 1−1.61iT−T2 1 - 1.61iT - T^{2}
37 1+T2 1 + T^{2}
41 1−0.618iT−T2 1 - 0.618iT - T^{2}
43 1+0.618T+T2 1 + 0.618T + T^{2}
47 1−T2 1 - T^{2}
53 1−1.61T+T2 1 - 1.61T + T^{2}
59 1−T2 1 - T^{2}
61 1−0.618iT−T2 1 - 0.618iT - T^{2}
67 1+1.61T+T2 1 + 1.61T + T^{2}
71 1+T2 1 + T^{2}
73 1+0.618iT−T2 1 + 0.618iT - T^{2}
79 1+T2 1 + T^{2}
83 1−T2 1 - T^{2}
89 1−T2 1 - T^{2}
97 1−1.61iT−T2 1 - 1.61iT - 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

−9.530149524256061558430184021143, −8.717308286078761966530515168776, −7.961484909077128434232190527719, −7.19871615981212674441257852204, −6.66056992543592197743923205830, −5.75916226514328112613533708924, −4.70230389213175719974731371294, −3.55150957332832544268050355362, −2.59033439286008265425169723469, −1.55622668146047121430898759567, 0.68141584551803872832964830496, 1.66763483875259198941601059458, 3.75714116110050099713019685473, 4.32181485230256221597820058648, 4.83839448092044101687631674856, 5.70183396207605482490342239316, 7.12391583944922861314067737597, 7.80932133243437555492443130164, 8.548110215490307501051792067808, 9.186166161128538447192956995948

Graph of the ZZ-function along the critical line