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Properties

Label 2-2023-119.104-c0-0-9
Degree $2$
Conductor $2023$
Sign $-0.730 - 0.682i$
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  + (−0.707 − 0.707i)2-s + (−0.382 + 0.923i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (−1.84 − 0.765i)11-s + (0.923 − 0.382i)14-s + 1.00·16-s − 1.00·18-s + (0.765 + 1.84i)22-s + (−0.923 − 0.382i)23-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)29-s + (−0.923 + 0.382i)37-s + (−0.707 + 0.707i)43-s + (0.382 + 0.923i)46-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.382 + 0.923i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (−1.84 − 0.765i)11-s + (0.923 − 0.382i)14-s + 1.00·16-s − 1.00·18-s + (0.765 + 1.84i)22-s + (−0.923 − 0.382i)23-s + (−0.707 + 0.707i)25-s + (−0.382 − 0.923i)29-s + (−0.923 + 0.382i)37-s + (−0.707 + 0.707i)43-s + (0.382 + 0.923i)46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $-0.730 - 0.682i$
Analytic conductor: \(1.00960\)
Root analytic conductor: \(1.00479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2023} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2023,\ (\ :0),\ -0.730 - 0.682i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1130444433\)
\(L(\frac12)\) \(\approx\) \(0.1130444433\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.382 - 0.923i)T \)
17 \( 1 \)
good2 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
3 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (1.84 + 0.765i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (0.923 + 0.382i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.382 + 0.923i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.923 - 0.382i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 + (0.923 - 0.382i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.923 - 0.382i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.707 - 0.707i)T^{2} \)
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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

−8.964367324665914290269330729965, −8.303745224401215828387510733895, −7.58920050227390773375753379092, −6.27586789103013294885941987729, −5.76296550009323824568937104300, −4.92562927172573978194030309624, −3.48480825052303940847392261706, −2.68948825320039007210869759856, −1.76306522237062423065164657053, −0.094684850389200964215648398358, 1.90568752594995315434336159958, 3.15734174510113795939262051237, 4.18268347937274210425010588851, 5.06387750899539912592618922430, 6.07360504695233137557957521897, 7.10139858447623713977789111354, 7.54943002111047537994533604963, 7.939699501997660525027045541593, 8.906978854493052913390362856015, 10.03614624998427500979083552879

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