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Properties

Label 2-2023-7.6-c0-0-0
Degree $2$
Conductor $2023$
Sign $-i$
Analytic cond. $1.00960$
Root an. cond. $1.00479$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $-i$
Analytic conductor: \(1.00960\)
Root analytic conductor: \(1.00479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2023} (1735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2023,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6850736554\)
\(L(\frac12)\) \(\approx\) \(0.6850736554\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - iT \)
17 \( 1 \)
good2 \( 1 + 0.618T + T^{2} \)
3 \( 1 + 0.618iT - T^{2} \)
5 \( 1 - 1.61iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.61iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 0.618iT - T^{2} \)
43 \( 1 + 0.618T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.61T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.618iT - T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 0.618iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.61iT - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line