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Properties

Label 2-2023-119.104-c0-0-7
Degree $2$
Conductor $2023$
Sign $0.950 - 0.311i$
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  + (1.14 + 1.14i)2-s + (−0.619 − 1.49i)3-s + 1.61i·4-s + (0.570 − 0.236i)5-s + (1.00 − 2.41i)6-s + (0.923 + 0.382i)7-s + (−0.707 + 0.707i)8-s + (−1.14 + 1.14i)9-s + (0.923 + 0.382i)10-s + (2.41 − 1.00i)12-s + (0.619 + 1.49i)14-s + (−0.707 − 0.707i)15-s − 2.61·18-s + (0.382 + 0.923i)20-s − 1.61i·21-s + ⋯
L(s)  = 1  + (1.14 + 1.14i)2-s + (−0.619 − 1.49i)3-s + 1.61i·4-s + (0.570 − 0.236i)5-s + (1.00 − 2.41i)6-s + (0.923 + 0.382i)7-s + (−0.707 + 0.707i)8-s + (−1.14 + 1.14i)9-s + (0.923 + 0.382i)10-s + (2.41 − 1.00i)12-s + (0.619 + 1.49i)14-s + (−0.707 − 0.707i)15-s − 2.61·18-s + (0.382 + 0.923i)20-s − 1.61i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $0.950 - 0.311i$
Analytic conductor: \(1.00960\)
Root analytic conductor: \(1.00479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2023} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2023,\ (\ :0),\ 0.950 - 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.071731250\)
\(L(\frac12)\) \(\approx\) \(2.071731250\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 \)
good2 \( 1 + (-1.14 - 1.14i)T + iT^{2} \)
3 \( 1 + (0.619 + 1.49i)T + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.570 + 0.236i)T + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (0.236 + 0.570i)T + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (1.49 + 0.619i)T + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (-1.14 + 1.14i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.437 + 0.437i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1.49 - 0.619i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
71 \( 1 + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (1.49 - 0.619i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.570 - 0.236i)T + (0.707 - 0.707i)T^{2} \)
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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

−8.913192479064282015573797994002, −8.164125682051386720794932698082, −7.45548958323471617621637266338, −6.94609776800499056245272419593, −6.06330592733006214036956397121, −5.56499416047299701363034549685, −5.04343934729599666711732835008, −3.91280853343450327866441820185, −2.40127464653199000919777804610, −1.43735911759551115120097344838, 1.54131376722925450266988326317, 2.73256871117177292011514598061, 3.70231861418432194114665326447, 4.39435050125401813180732339237, 4.98582411332358257082009486585, 5.60330944336906368109941406688, 6.42286598128106570350222928008, 7.80680493695491968174432287533, 8.904805758545097501502156476238, 9.858153975496469796493141154342

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