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Properties

Label 2-2023-119.13-c0-0-6
Degree $2$
Conductor $2023$
Sign $0.877 - 0.479i$
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  − 0.347i·2-s + 0.879·4-s + (−0.707 + 0.707i)7-s − 0.652i·8-s + i·9-s + (−0.707 + 0.707i)11-s + (0.245 + 0.245i)14-s + 0.652·16-s + 0.347·18-s + (0.245 + 0.245i)22-s + (1.08 − 1.08i)23-s + i·25-s + (−0.621 + 0.621i)28-s + (1.32 + 1.32i)29-s − 0.879i·32-s + ⋯
L(s)  = 1  − 0.347i·2-s + 0.879·4-s + (−0.707 + 0.707i)7-s − 0.652i·8-s + i·9-s + (−0.707 + 0.707i)11-s + (0.245 + 0.245i)14-s + 0.652·16-s + 0.347·18-s + (0.245 + 0.245i)22-s + (1.08 − 1.08i)23-s + i·25-s + (−0.621 + 0.621i)28-s + (1.32 + 1.32i)29-s − 0.879i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $0.877 - 0.479i$
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: \((2,\ 2023,\ (\ :0),\ 0.877 - 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.295979485\)
\(L(\frac12)\) \(\approx\) \(1.295979485\)
\(L(1)\) 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 \( 1 + 0.347iT - T^{2} \)
3 \( 1 - iT^{2} \)
5 \( 1 - iT^{2} \)
11 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1.08 + 1.08i)T - iT^{2} \)
29 \( 1 + (-1.32 - 1.32i)T + iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-0.245 - 0.245i)T + iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 - 1.53iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.53iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + (1.32 + 1.32i)T + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (-0.245 + 0.245i)T - iT^{2} \)
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

−9.528434024640815106977260466151, −8.630016278366689486468468609901, −7.77521283073647779927805132955, −7.00163704151670213406207053966, −6.37867743110597634122932799928, −5.33753941590993557999752696618, −4.66175836031676004533278589547, −3.12112835386395640415769366253, −2.67756901666978746703545051226, −1.63191556868238318205774277902, 0.953137792944884991110089363722, 2.62619718943693783697571415978, 3.28734201889201665703930898541, 4.30810382600953414393692799904, 5.61543788362505969890568840916, 6.18213484376763254227250842985, 6.91157955277770726027438314790, 7.53965677138123453679676540375, 8.389943985406621185303005594093, 9.237424660354513461839620731050

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