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Properties

Label 2-2023-119.118-c0-0-5
Degree $2$
Conductor $2023$
Sign $0.985 - 0.168i$
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  + 1.87·2-s + 2.53·4-s + i·7-s + 2.87·8-s − 9-s − i·11-s + 1.87i·14-s + 2.87·16-s − 1.87·18-s − 1.87i·22-s + 0.347i·23-s − 25-s + 2.53i·28-s − 1.53i·29-s + 2.53·32-s + ⋯
L(s)  = 1  + 1.87·2-s + 2.53·4-s + i·7-s + 2.87·8-s − 9-s − i·11-s + 1.87i·14-s + 2.87·16-s − 1.87·18-s − 1.87i·22-s + 0.347i·23-s − 25-s + 2.53i·28-s − 1.53i·29-s + 2.53·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $0.985 - 0.168i$
Analytic conductor: \(1.00960\)
Root analytic conductor: \(1.00479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2023} (2022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2023,\ (\ :0),\ 0.985 - 0.168i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.360864354\)
\(L(\frac12)\) \(\approx\) \(3.360864354\)
\(L(1)\) 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 \( 1 - 1.87T + T^{2} \)
3 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 0.347iT - T^{2} \)
29 \( 1 + 1.53iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.87iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 0.347T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.347T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + 1.53iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.87iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.341433207329269568964256614734, −8.334540299786406446834797076859, −7.71766235741264537678956118527, −6.33631824204349189040774277321, −6.07794564210279202382627885484, −5.38790721755357517239999239181, −4.59489165940278326519231684566, −3.45363347518319274463987677254, −2.90888091936582648818679529265, −1.97384257744188545655507611630, 1.77553930961386423544167316422, 2.80298199850150122553010819001, 3.72585410406036103302416776661, 4.36931129449905530392264372414, 5.23010658630410797722831402534, 5.88759656019654461225263406164, 6.87735096307785968777502548458, 7.29021329014677346175589790674, 8.229296161952321189740779591326, 9.476997217357531234774648777078

Graph of the $Z$-function along the critical line