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Properties

Label 2-2023-119.111-c0-0-7
Degree $2$
Conductor $2023$
Sign $0.920 - 0.390i$
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  + (−1.32 + 1.32i)2-s − 2.53i·4-s + (−0.382 − 0.923i)7-s + (2.03 + 2.03i)8-s + (0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s + (1.73 + 0.719i)14-s − 2.87·16-s − 1.87·18-s + (−0.719 + 1.73i)22-s + (0.320 − 0.132i)23-s + (−0.707 − 0.707i)25-s + (−2.33 + 0.968i)28-s + (0.586 − 1.41i)29-s + (1.79 − 1.79i)32-s + ⋯
L(s)  = 1  + (−1.32 + 1.32i)2-s − 2.53i·4-s + (−0.382 − 0.923i)7-s + (2.03 + 2.03i)8-s + (0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s + (1.73 + 0.719i)14-s − 2.87·16-s − 1.87·18-s + (−0.719 + 1.73i)22-s + (0.320 − 0.132i)23-s + (−0.707 − 0.707i)25-s + (−2.33 + 0.968i)28-s + (0.586 − 1.41i)29-s + (1.79 − 1.79i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $0.920 - 0.390i$
Analytic conductor: \(1.00960\)
Root analytic conductor: \(1.00479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2023} (468, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2023,\ (\ :0),\ 0.920 - 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5907810550\)
\(L(\frac12)\) \(\approx\) \(0.5907810550\)
\(L(1)\) 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 \( 1 + (1.32 - 1.32i)T - iT^{2} \)
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 \( 1 + (-0.320 + 0.132i)T + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.586 + 1.41i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (1.73 + 0.719i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.245 - 0.245i)T + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.245 + 0.245i)T - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (0.707 - 0.707i)T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + (-1.41 - 0.586i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (-1.73 + 0.719i)T + (0.707 - 0.707i)T^{2} \)
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

−9.365705217760419819543620475594, −8.395643790409178577969740809078, −7.87406417571929670327279084982, −7.04255074982890966835119319535, −6.61636769309801718460513290551, −5.78924019869823806696488860227, −4.75245207070011123284097132614, −3.85282703268918039514955837117, −1.97921674463162972425889453184, −0.78270009970735484132637436208, 1.24146955598028235168051050265, 2.09725347212086789237387670850, 3.29103461228031498622570831054, 3.80026800149895616372664764763, 5.09040054372738839943137262983, 6.55026360339042355733429261078, 7.06881197147832145187184071819, 8.095545901946871630173927902326, 8.989794835954454599463867761689, 9.235045371158740712158712389540

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