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Properties

Label 2-2023-119.104-c0-0-4
Degree $2$
Conductor $2023$
Sign $-0.186 - 0.982i$
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  + (1.08 + 1.08i)2-s + 1.34i·4-s + (−0.382 + 0.923i)7-s + (−0.376 + 0.376i)8-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s + (−1.41 + 0.586i)14-s + 0.532·16-s + 1.53·18-s + (0.586 + 1.41i)22-s + (−1.73 − 0.719i)23-s + (−0.707 + 0.707i)25-s + (−1.24 − 0.515i)28-s + (0.132 + 0.320i)29-s + (0.952 + 0.952i)32-s + ⋯
L(s)  = 1  + (1.08 + 1.08i)2-s + 1.34i·4-s + (−0.382 + 0.923i)7-s + (−0.376 + 0.376i)8-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s + (−1.41 + 0.586i)14-s + 0.532·16-s + 1.53·18-s + (0.586 + 1.41i)22-s + (−1.73 − 0.719i)23-s + (−0.707 + 0.707i)25-s + (−1.24 − 0.515i)28-s + (0.132 + 0.320i)29-s + (0.952 + 0.952i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $-0.186 - 0.982i$
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: \((2,\ 2023,\ (\ :0),\ -0.186 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.147871118\)
\(L(\frac12)\) \(\approx\) \(2.147871118\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.382 - 0.923i)T \)
17 \( 1 \)
good2 \( 1 + (-1.08 - 1.08i)T + iT^{2} \)
3 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.923 - 0.382i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (1.73 + 0.719i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.132 - 0.320i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (-1.41 + 0.586i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (1.32 - 1.32i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.32 + 1.32i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + (-0.320 + 0.132i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (1.41 + 0.586i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.707 - 0.707i)T^{2} \)
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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

−9.711792940224005843079172193785, −8.548403147174749513819566666753, −7.78869733520262049206951593076, −6.85532622346666756362262761470, −6.31342460937007466366435979644, −5.80132597399867350899154273996, −4.71944426559374516759203525251, −4.03834692142669286273972239661, −3.24674028877240444764833526414, −1.77696260423594928211764806090, 1.31268871220544732284513677528, 2.25598710434562917524985921055, 3.49117669867555144562170907563, 4.07262355933729387056809207943, 4.65359975590132740585492031177, 5.79752640319132108985246029514, 6.50746724980707729105463670923, 7.58213336200464424627057811062, 8.210579333197365654096125159427, 9.638578260205761644020545500505

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