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Properties

Label 2-2023-119.13-c0-0-4
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

\(L(\frac{1}{2})\) \(\approx\) \(1.247480126\)
\(L(\frac12)\) \(\approx\) \(1.247480126\)
\(L(1)\) 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

−9.576445465832761151403738625300, −8.395198526202575403389668764255, −7.80249630130476733945115327099, −7.39214467620720805035022254317, −6.74889692876833320031348312031, −5.44291887930623831464983970255, −5.05368005490257214261260137528, −4.29162603129834203771891992828, −2.62427933590085173686142886029, −1.86819339684400054043210282572, 0.890365777496651126933968140306, 2.23130068785396647242236317922, 3.01990719995001125974855017657, 3.74814502337373902439604130866, 5.02750399244484743580869646376, 5.79845708789402628857306386107, 6.59192676217316787645099105761, 7.68380396139913703914153000182, 8.404082016578424704872088555805, 9.194710291321763381138905314214

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