Properties

Label 2-2023-1.1-c3-0-350
Degree $2$
Conductor $2023$
Sign $-1$
Analytic cond. $119.360$
Root an. cond. $10.9252$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 6·3-s − 7·4-s + 20·5-s − 6·6-s + 7·7-s + 15·8-s + 9·9-s − 20·10-s − 60·11-s − 42·12-s − 68·13-s − 7·14-s + 120·15-s + 41·16-s − 9·18-s − 70·19-s − 140·20-s + 42·21-s + 60·22-s + 176·23-s + 90·24-s + 275·25-s + 68·26-s − 108·27-s − 49·28-s + 90·29-s + ⋯
L(s)  = 1  − 0.353·2-s + 1.15·3-s − 7/8·4-s + 1.78·5-s − 0.408·6-s + 0.377·7-s + 0.662·8-s + 1/3·9-s − 0.632·10-s − 1.64·11-s − 1.01·12-s − 1.45·13-s − 0.133·14-s + 2.06·15-s + 0.640·16-s − 0.117·18-s − 0.845·19-s − 1.56·20-s + 0.436·21-s + 0.581·22-s + 1.59·23-s + 0.765·24-s + 11/5·25-s + 0.512·26-s − 0.769·27-s − 0.330·28-s + 0.576·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(119.360\)
Root analytic conductor: \(10.9252\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2023} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2023,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - p T \)
17 \( 1 \)
good2 \( 1 + T + p^{3} T^{2} \)
3 \( 1 - 2 p T + p^{3} T^{2} \)
5 \( 1 - 4 p T + p^{3} T^{2} \)
11 \( 1 + 60 T + p^{3} T^{2} \)
13 \( 1 + 68 T + p^{3} T^{2} \)
19 \( 1 + 70 T + p^{3} T^{2} \)
23 \( 1 - 176 T + p^{3} T^{2} \)
29 \( 1 - 90 T + p^{3} T^{2} \)
31 \( 1 + 196 T + p^{3} T^{2} \)
37 \( 1 + 22 T + p^{3} T^{2} \)
41 \( 1 - 138 T + p^{3} T^{2} \)
43 \( 1 - 328 T + p^{3} T^{2} \)
47 \( 1 + 12 T + p^{3} T^{2} \)
53 \( 1 + 234 T + p^{3} T^{2} \)
59 \( 1 + 54 T + p^{3} T^{2} \)
61 \( 1 + 44 T + p^{3} T^{2} \)
67 \( 1 + 596 T + p^{3} T^{2} \)
71 \( 1 + 200 T + p^{3} T^{2} \)
73 \( 1 + 1122 T + p^{3} T^{2} \)
79 \( 1 + 480 T + p^{3} T^{2} \)
83 \( 1 + 838 T + p^{3} T^{2} \)
89 \( 1 - 778 T + p^{3} T^{2} \)
97 \( 1 + 1142 T + p^{3} T^{2} \)
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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

−8.672860537990974570564801969137, −7.77461277858148526791521862739, −7.17303226021949612561219220912, −5.75587429591641957354222486764, −5.18245069940158503532695466139, −4.49240016796289910963663203287, −2.87808099694703156613029811080, −2.47956246608874700658919385076, −1.50417645399165074806134334412, 0, 1.50417645399165074806134334412, 2.47956246608874700658919385076, 2.87808099694703156613029811080, 4.49240016796289910963663203287, 5.18245069940158503532695466139, 5.75587429591641957354222486764, 7.17303226021949612561219220912, 7.77461277858148526791521862739, 8.672860537990974570564801969137

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