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Properties

Label 2-2023-119.55-c0-0-0
Degree $2$
Conductor $2023$
Sign $0.992 - 0.122i$
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  + 1.61i·2-s + (−1.14 − 1.14i)3-s − 1.61·4-s + (−0.437 − 0.437i)5-s + (1.85 − 1.85i)6-s + (−0.707 + 0.707i)7-s − i·8-s + 1.61i·9-s + (0.707 − 0.707i)10-s + (1.85 + 1.85i)12-s + (−1.14 − 1.14i)14-s + i·15-s − 2.61·18-s + (0.707 + 0.707i)20-s + 1.61·21-s + ⋯
L(s)  = 1  + 1.61i·2-s + (−1.14 − 1.14i)3-s − 1.61·4-s + (−0.437 − 0.437i)5-s + (1.85 − 1.85i)6-s + (−0.707 + 0.707i)7-s − i·8-s + 1.61i·9-s + (0.707 − 0.707i)10-s + (1.85 + 1.85i)12-s + (−1.14 − 1.14i)14-s + i·15-s − 2.61·18-s + (0.707 + 0.707i)20-s + 1.61·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $0.992 - 0.122i$
Analytic conductor: \(1.00960\)
Root analytic conductor: \(1.00479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2023} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2023,\ (\ :0),\ 0.992 - 0.122i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4520894738\)
\(L(\frac12)\) \(\approx\) \(0.4520894738\)
\(L(1)\) 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 \( 1 - 1.61iT - T^{2} \)
3 \( 1 + (1.14 + 1.14i)T + iT^{2} \)
5 \( 1 + (0.437 + 0.437i)T + iT^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + (-0.437 - 0.437i)T + iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-1.14 + 1.14i)T - iT^{2} \)
43 \( 1 + 1.61iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.618iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-1.14 + 1.14i)T - iT^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-1.14 - 1.14i)T + iT^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.437 + 0.437i)T + iT^{2} \)
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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

−8.894742280715298548245562019987, −8.305804088458522932529580275868, −7.54651347142376813457225848679, −6.78887927288040637699320698862, −6.40467960669621724530618150879, −5.53194969323156009347147069514, −5.16715527446885008738274431463, −3.97180508741669060209206749229, −2.26096067137870408186824678075, −0.49522163476008788606799979424, 0.970006550425115842429392189670, 2.74540193565417070615994590493, 3.58166474914910206789967502174, 4.18447214057775579102974088405, 4.85523692782904740676600781328, 5.97831751284781130858042607953, 6.78109129215182927044422785852, 7.87558562972693190929281606011, 9.248581156060696811196213749652, 9.649371365820437167024254511357

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