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Properties

Label 2-2023-119.118-c0-0-3
Degree $2$
Conductor $2023$
Sign $0.410 + 0.911i$
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  − 1.53·2-s + 1.34·4-s + i·7-s − 0.532·8-s − 9-s − i·11-s − 1.53i·14-s − 0.532·16-s + 1.53·18-s + 1.53i·22-s − 1.87i·23-s − 25-s + 1.34i·28-s − 0.347i·29-s + 1.34·32-s + ⋯
L(s)  = 1  − 1.53·2-s + 1.34·4-s + i·7-s − 0.532·8-s − 9-s − i·11-s − 1.53i·14-s − 0.532·16-s + 1.53·18-s + 1.53i·22-s − 1.87i·23-s − 25-s + 1.34i·28-s − 0.347i·29-s + 1.34·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $0.410 + 0.911i$
Analytic conductor: \(1.00960\)
Root analytic conductor: \(1.00479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2023} (2022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2023,\ (\ :0),\ 0.410 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3840641073\)
\(L(\frac12)\) \(\approx\) \(0.3840641073\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - iT \)
17 \( 1 \)
good2 \( 1 + 1.53T + T^{2} \)
3 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.87iT - T^{2} \)
29 \( 1 + 0.347iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 1.53iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.87T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.87T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + 0.347iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.53iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.012559155119754646807027707724, −8.532832333607967656260370230588, −8.068473483121951445617456962171, −7.07282892232719285744953670715, −6.03283883601741273295377449881, −5.63249123993757475067413102078, −4.21694942121274496795247682812, −2.80438526837903144177436316749, −2.17972705380269107249164440375, −0.50520965818687737848697960532, 1.18655545695114325122636657150, 2.26410997869816283577827879705, 3.53630148874755540983525967819, 4.59092515862668573778498521511, 5.69878736270048725278712788943, 6.70566399828813917297009893091, 7.57213833493246970661989920368, 7.76771310776585383755212617436, 8.847201325608735977525591797187, 9.418485561582072036302671788812

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