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Properties

Label 2-2023-119.13-c0-0-5
Degree $2$
Conductor $2023$
Sign $-0.615 - 0.788i$
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  + i·2-s + (−0.707 + 0.707i)7-s + i·8-s + i·9-s + (1.41 − 1.41i)11-s + (−0.707 − 0.707i)14-s − 16-s − 18-s + (1.41 + 1.41i)22-s + (−0.707 + 0.707i)23-s + i·25-s + (0.707 + 0.707i)29-s + (−0.707 − 0.707i)37-s − i·43-s + (−0.707 − 0.707i)46-s + ⋯
L(s)  = 1  + i·2-s + (−0.707 + 0.707i)7-s + i·8-s + i·9-s + (1.41 − 1.41i)11-s + (−0.707 − 0.707i)14-s − 16-s − 18-s + (1.41 + 1.41i)22-s + (−0.707 + 0.707i)23-s + i·25-s + (0.707 + 0.707i)29-s + (−0.707 − 0.707i)37-s − i·43-s + (−0.707 − 0.707i)46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $-0.615 - 0.788i$
Analytic conductor: \(1.00960\)
Root analytic conductor: \(1.00479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2023} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2023,\ (\ :0),\ -0.615 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.242565390\)
\(L(\frac12)\) \(\approx\) \(1.242565390\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 \)
good2 \( 1 - iT - T^{2} \)
3 \( 1 - iT^{2} \)
5 \( 1 - iT^{2} \)
11 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.194656989492731624683443480906, −8.766372761664506298005663737307, −8.029572325652036146221174092132, −7.13279120410163787868057816342, −6.47226960896088648324614913100, −5.71894379287573250430348992278, −5.26505731556590713485426035408, −3.86090685877460714262759605142, −2.93147367209297020361482554001, −1.74250258170730366775253582787, 0.930930441476495510441877065770, 2.05096787282279998136053187784, 3.16631162856260041484953541636, 4.02526410295277897126651948742, 4.45884273032456029660855638658, 6.33624267786232214147435302775, 6.55915726290957865652173759274, 7.26936881785176742011699450100, 8.498316489211341332460125601865, 9.517214914206953688909902389295

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