Properties

Label 2-23-23.2-c3-0-1
Degree $2$
Conductor $23$
Sign $0.671 - 0.741i$
Analytic cond. $1.35704$
Root an. cond. $1.16492$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.91 + 1.23i)2-s + (0.776 + 5.40i)3-s + (−1.16 − 2.55i)4-s + (−0.907 − 0.266i)5-s + (−5.16 + 11.3i)6-s + (8.82 − 10.1i)7-s + (3.50 − 24.3i)8-s + (−2.69 + 0.790i)9-s + (−1.41 − 1.62i)10-s + (−44.9 + 28.8i)11-s + (12.8 − 8.28i)12-s + (−21.9 − 25.3i)13-s + (29.4 − 8.65i)14-s + (0.735 − 5.11i)15-s + (22.0 − 25.4i)16-s + (−14.2 + 31.2i)17-s + ⋯
L(s)  = 1  + (0.677 + 0.435i)2-s + (0.149 + 1.03i)3-s + (−0.145 − 0.318i)4-s + (−0.0812 − 0.0238i)5-s + (−0.351 + 0.770i)6-s + (0.476 − 0.550i)7-s + (0.154 − 1.07i)8-s + (−0.0997 + 0.0292i)9-s + (−0.0446 − 0.0515i)10-s + (−1.23 + 0.791i)11-s + (0.309 − 0.199i)12-s + (−0.468 − 0.541i)13-s + (0.562 − 0.165i)14-s + (0.0126 − 0.0880i)15-s + (0.344 − 0.397i)16-s + (−0.203 + 0.445i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.671 - 0.741i$
Analytic conductor: \(1.35704\)
Root analytic conductor: \(1.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :3/2),\ 0.671 - 0.741i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.36130 + 0.603714i\)
\(L(\frac12)\) \(\approx\) \(1.36130 + 0.603714i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-88.8 - 65.3i)T \)
good2 \( 1 + (-1.91 - 1.23i)T + (3.32 + 7.27i)T^{2} \)
3 \( 1 + (-0.776 - 5.40i)T + (-25.9 + 7.60i)T^{2} \)
5 \( 1 + (0.907 + 0.266i)T + (105. + 67.5i)T^{2} \)
7 \( 1 + (-8.82 + 10.1i)T + (-48.8 - 339. i)T^{2} \)
11 \( 1 + (44.9 - 28.8i)T + (552. - 1.21e3i)T^{2} \)
13 \( 1 + (21.9 + 25.3i)T + (-312. + 2.17e3i)T^{2} \)
17 \( 1 + (14.2 - 31.2i)T + (-3.21e3 - 3.71e3i)T^{2} \)
19 \( 1 + (-36.1 - 79.0i)T + (-4.49e3 + 5.18e3i)T^{2} \)
29 \( 1 + (-19.2 + 42.1i)T + (-1.59e4 - 1.84e4i)T^{2} \)
31 \( 1 + (-29.9 + 207. i)T + (-2.85e4 - 8.39e3i)T^{2} \)
37 \( 1 + (268. - 78.6i)T + (4.26e4 - 2.73e4i)T^{2} \)
41 \( 1 + (-315. - 92.7i)T + (5.79e4 + 3.72e4i)T^{2} \)
43 \( 1 + (48.6 + 338. i)T + (-7.62e4 + 2.23e4i)T^{2} \)
47 \( 1 - 460.T + 1.03e5T^{2} \)
53 \( 1 + (159. - 183. i)T + (-2.11e4 - 1.47e5i)T^{2} \)
59 \( 1 + (78.4 + 90.5i)T + (-2.92e4 + 2.03e5i)T^{2} \)
61 \( 1 + (-57.8 + 402. i)T + (-2.17e5 - 6.39e4i)T^{2} \)
67 \( 1 + (-732. - 470. i)T + (1.24e5 + 2.73e5i)T^{2} \)
71 \( 1 + (-241. - 155. i)T + (1.48e5 + 3.25e5i)T^{2} \)
73 \( 1 + (379. + 831. i)T + (-2.54e5 + 2.93e5i)T^{2} \)
79 \( 1 + (-516. - 595. i)T + (-7.01e4 + 4.88e5i)T^{2} \)
83 \( 1 + (1.29e3 - 381. i)T + (4.81e5 - 3.09e5i)T^{2} \)
89 \( 1 + (105. + 736. i)T + (-6.76e5 + 1.98e5i)T^{2} \)
97 \( 1 + (-479. - 140. i)T + (7.67e5 + 4.93e5i)T^{2} \)
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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

−17.29697537185432841707112662266, −15.69694376772737870468705050535, −15.24363671469252760158009300233, −14.02531521298387496015688098324, −12.71006266376815316926325078560, −10.54990249982423496562737260379, −9.768808743670697392230957846737, −7.57329911302301603393845055035, −5.35090018885185474126126455463, −4.14659688358253887156997723238, 2.58926151151444847432854054666, 5.09166933360828609623295624373, 7.36064352715927995335042758703, 8.646639268085941773992121489879, 11.12281920654267705144189524493, 12.27975886690477429599936335993, 13.25480191563905105324700492134, 14.14125503367680571539409953897, 15.85017867271983052651386568461, 17.55944402579243420474983237142

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