Properties

Label 2-23-23.7-c2-0-1
Degree $2$
Conductor $23$
Sign $0.914 - 0.403i$
Analytic cond. $0.626704$
Root an. cond. $0.791646$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.881 + 1.01i)2-s + (2.64 − 1.70i)3-s + (0.311 + 2.16i)4-s + (−2.08 + 0.951i)5-s + (−0.602 + 4.19i)6-s + (−2.90 − 9.89i)7-s + (−7.00 − 4.50i)8-s + (0.375 − 0.822i)9-s + (0.867 − 2.95i)10-s + (−6.07 + 5.26i)11-s + (4.51 + 5.20i)12-s + (14.4 + 4.25i)13-s + (12.6 + 5.76i)14-s + (−3.89 + 6.06i)15-s + (2.34 − 0.689i)16-s + (−13.1 − 1.89i)17-s + ⋯
L(s)  = 1  + (−0.440 + 0.508i)2-s + (0.882 − 0.567i)3-s + (0.0779 + 0.541i)4-s + (−0.416 + 0.190i)5-s + (−0.100 + 0.698i)6-s + (−0.415 − 1.41i)7-s + (−0.875 − 0.562i)8-s + (0.0417 − 0.0913i)9-s + (0.0867 − 0.295i)10-s + (−0.552 + 0.478i)11-s + (0.376 + 0.434i)12-s + (1.11 + 0.327i)13-s + (0.901 + 0.411i)14-s + (−0.259 + 0.404i)15-s + (0.146 − 0.0430i)16-s + (−0.774 − 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.914 - 0.403i$
Analytic conductor: \(0.626704\)
Root analytic conductor: \(0.791646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :1),\ 0.914 - 0.403i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.841690 + 0.177475i\)
\(L(\frac12)\) \(\approx\) \(0.841690 + 0.177475i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-22.1 - 6.20i)T \)
good2 \( 1 + (0.881 - 1.01i)T + (-0.569 - 3.95i)T^{2} \)
3 \( 1 + (-2.64 + 1.70i)T + (3.73 - 8.18i)T^{2} \)
5 \( 1 + (2.08 - 0.951i)T + (16.3 - 18.8i)T^{2} \)
7 \( 1 + (2.90 + 9.89i)T + (-41.2 + 26.4i)T^{2} \)
11 \( 1 + (6.07 - 5.26i)T + (17.2 - 119. i)T^{2} \)
13 \( 1 + (-14.4 - 4.25i)T + (142. + 91.3i)T^{2} \)
17 \( 1 + (13.1 + 1.89i)T + (277. + 81.4i)T^{2} \)
19 \( 1 + (-26.2 + 3.77i)T + (346. - 101. i)T^{2} \)
29 \( 1 + (-0.753 + 5.23i)T + (-806. - 236. i)T^{2} \)
31 \( 1 + (29.7 + 19.1i)T + (399. + 874. i)T^{2} \)
37 \( 1 + (32.5 + 14.8i)T + (896. + 1.03e3i)T^{2} \)
41 \( 1 + (-7.60 - 16.6i)T + (-1.10e3 + 1.27e3i)T^{2} \)
43 \( 1 + (-8.28 - 12.8i)T + (-768. + 1.68e3i)T^{2} \)
47 \( 1 - 72.6T + 2.20e3T^{2} \)
53 \( 1 + (25.8 + 88.1i)T + (-2.36e3 + 1.51e3i)T^{2} \)
59 \( 1 + (71.1 + 20.8i)T + (2.92e3 + 1.88e3i)T^{2} \)
61 \( 1 + (15.2 - 23.7i)T + (-1.54e3 - 3.38e3i)T^{2} \)
67 \( 1 + (5.29 + 4.58i)T + (638. + 4.44e3i)T^{2} \)
71 \( 1 + (-18.7 + 21.6i)T + (-717. - 4.98e3i)T^{2} \)
73 \( 1 + (1.65 + 11.5i)T + (-5.11e3 + 1.50e3i)T^{2} \)
79 \( 1 + (-5.91 + 20.1i)T + (-5.25e3 - 3.37e3i)T^{2} \)
83 \( 1 + (15.0 + 6.85i)T + (4.51e3 + 5.20e3i)T^{2} \)
89 \( 1 + (-21.3 - 33.1i)T + (-3.29e3 + 7.20e3i)T^{2} \)
97 \( 1 + (77.7 - 35.4i)T + (6.16e3 - 7.11e3i)T^{2} \)
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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

−17.75136145122288937999373352113, −16.49341297516718847727031548066, −15.47852907258902155995284640000, −13.76026400007671495055827154018, −13.04023352838451977902926988399, −11.12369738934736905519692447788, −9.193506155720072687808625249177, −7.71460660646040371062899645238, −7.07885428875466974222132406428, −3.50959259140113260326206463056, 2.99897238130722074075805631366, 5.74137923051203711925457380547, 8.606221946965487034964007726030, 9.231891547025923017877981442687, 10.79033567244261862943439898790, 12.17367779977486827364746689328, 13.95831138836016025728756514535, 15.50706724580438379338418176615, 15.66643909892620264905496319087, 18.14287690664937343191931716911

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