Properties

Label 2-2023-1.1-c3-0-220
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 + 2·3-s − 7·4-s − 16·5-s − 2·6-s + 7·7-s + 15·8-s − 23·9-s + 16·10-s + 8·11-s − 14·12-s + 28·13-s − 7·14-s − 32·15-s + 41·16-s + 23·18-s − 110·19-s + 112·20-s + 14·21-s − 8·22-s − 48·23-s + 30·24-s + 131·25-s − 28·26-s − 100·27-s − 49·28-s + 110·29-s + ⋯
L(s)  = 1  − 0.353·2-s + 0.384·3-s − 7/8·4-s − 1.43·5-s − 0.136·6-s + 0.377·7-s + 0.662·8-s − 0.851·9-s + 0.505·10-s + 0.219·11-s − 0.336·12-s + 0.597·13-s − 0.133·14-s − 0.550·15-s + 0.640·16-s + 0.301·18-s − 1.32·19-s + 1.25·20-s + 0.145·21-s − 0.0775·22-s − 0.435·23-s + 0.255·24-s + 1.04·25-s − 0.211·26-s − 0.712·27-s − 0.330·28-s + 0.704·29-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 \( 1 - 2 T + p^{3} T^{2} \)
5 \( 1 + 16 T + p^{3} T^{2} \)
11 \( 1 - 8 T + p^{3} T^{2} \)
13 \( 1 - 28 T + p^{3} T^{2} \)
19 \( 1 + 110 T + p^{3} T^{2} \)
23 \( 1 + 48 T + p^{3} T^{2} \)
29 \( 1 - 110 T + p^{3} T^{2} \)
31 \( 1 + 12 T + p^{3} T^{2} \)
37 \( 1 - 246 T + p^{3} T^{2} \)
41 \( 1 + 182 T + p^{3} T^{2} \)
43 \( 1 - 128 T + p^{3} T^{2} \)
47 \( 1 - 324 T + p^{3} T^{2} \)
53 \( 1 + 162 T + p^{3} T^{2} \)
59 \( 1 - 810 T + p^{3} T^{2} \)
61 \( 1 - 8 p T + p^{3} T^{2} \)
67 \( 1 - 244 T + p^{3} T^{2} \)
71 \( 1 - 768 T + p^{3} T^{2} \)
73 \( 1 - 702 T + p^{3} T^{2} \)
79 \( 1 + 440 T + p^{3} T^{2} \)
83 \( 1 + 1302 T + p^{3} T^{2} \)
89 \( 1 - 730 T + p^{3} T^{2} \)
97 \( 1 + 294 T + p^{3} T^{2} \)
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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

−8.281089090805232835847017922864, −8.119469996871084004080088466547, −7.09348779906070825792719433088, −6.01567260665594557998558968020, −4.97653078235414337154472409882, −4.06109927005786865888711679968, −3.70045820096996972130890872822, −2.40758285035892424150129739254, −0.917578334401803458965068675174, 0, 0.917578334401803458965068675174, 2.40758285035892424150129739254, 3.70045820096996972130890872822, 4.06109927005786865888711679968, 4.97653078235414337154472409882, 6.01567260665594557998558968020, 7.09348779906070825792719433088, 8.119469996871084004080088466547, 8.281089090805232835847017922864

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