Properties

Label 2-23-23.4-c3-0-0
Degree $2$
Conductor $23$
Sign $-0.296 - 0.955i$
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  + (1.49 + 3.28i)2-s + (−7.64 + 2.24i)3-s + (−3.28 + 3.78i)4-s + (15.0 + 9.67i)5-s + (−18.8 − 21.7i)6-s + (−2.56 − 17.8i)7-s + (10.3 + 3.03i)8-s + (30.6 − 19.7i)9-s + (−9.18 + 63.8i)10-s + (−10.3 + 22.7i)11-s + (16.5 − 36.3i)12-s + (7.78 − 54.1i)13-s + (54.6 − 35.1i)14-s + (−136. − 40.1i)15-s + (11.2 + 78.1i)16-s + (−35.0 − 40.4i)17-s + ⋯
L(s)  = 1  + (0.529 + 1.16i)2-s + (−1.47 + 0.431i)3-s + (−0.410 + 0.473i)4-s + (1.34 + 0.864i)5-s + (−1.28 − 1.47i)6-s + (−0.138 − 0.961i)7-s + (0.457 + 0.134i)8-s + (1.13 − 0.729i)9-s + (−0.290 + 2.01i)10-s + (−0.284 + 0.623i)11-s + (0.398 − 0.873i)12-s + (0.166 − 1.15i)13-s + (1.04 − 0.670i)14-s + (−2.35 − 0.690i)15-s + (0.175 + 1.22i)16-s + (−0.500 − 0.577i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.296 - 0.955i$
Analytic conductor: \(1.35704\)
Root analytic conductor: \(1.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :3/2),\ -0.296 - 0.955i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.687275 + 0.932953i\)
\(L(\frac12)\) \(\approx\) \(0.687275 + 0.932953i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-26.7 + 107. i)T \)
good2 \( 1 + (-1.49 - 3.28i)T + (-5.23 + 6.04i)T^{2} \)
3 \( 1 + (7.64 - 2.24i)T + (22.7 - 14.5i)T^{2} \)
5 \( 1 + (-15.0 - 9.67i)T + (51.9 + 113. i)T^{2} \)
7 \( 1 + (2.56 + 17.8i)T + (-329. + 96.6i)T^{2} \)
11 \( 1 + (10.3 - 22.7i)T + (-871. - 1.00e3i)T^{2} \)
13 \( 1 + (-7.78 + 54.1i)T + (-2.10e3 - 618. i)T^{2} \)
17 \( 1 + (35.0 + 40.4i)T + (-699. + 4.86e3i)T^{2} \)
19 \( 1 + (11.3 - 13.0i)T + (-976. - 6.78e3i)T^{2} \)
29 \( 1 + (-55.4 - 64.0i)T + (-3.47e3 + 2.41e4i)T^{2} \)
31 \( 1 + (100. + 29.3i)T + (2.50e4 + 1.61e4i)T^{2} \)
37 \( 1 + (160. - 102. i)T + (2.10e4 - 4.60e4i)T^{2} \)
41 \( 1 + (93.7 + 60.2i)T + (2.86e4 + 6.26e4i)T^{2} \)
43 \( 1 + (-279. + 82.0i)T + (6.68e4 - 4.29e4i)T^{2} \)
47 \( 1 + 504.T + 1.03e5T^{2} \)
53 \( 1 + (-7.76 - 54.0i)T + (-1.42e5 + 4.19e4i)T^{2} \)
59 \( 1 + (42.9 - 298. i)T + (-1.97e5 - 5.78e4i)T^{2} \)
61 \( 1 + (800. + 235. i)T + (1.90e5 + 1.22e5i)T^{2} \)
67 \( 1 + (1.95 + 4.28i)T + (-1.96e5 + 2.27e5i)T^{2} \)
71 \( 1 + (-382. - 838. i)T + (-2.34e5 + 2.70e5i)T^{2} \)
73 \( 1 + (165. - 190. i)T + (-5.53e4 - 3.85e5i)T^{2} \)
79 \( 1 + (-35.5 + 247. i)T + (-4.73e5 - 1.38e5i)T^{2} \)
83 \( 1 + (-828. + 532. i)T + (2.37e5 - 5.20e5i)T^{2} \)
89 \( 1 + (-140. + 41.3i)T + (5.93e5 - 3.81e5i)T^{2} \)
97 \( 1 + (-61.1 - 39.3i)T + (3.79e5 + 8.30e5i)T^{2} \)
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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

−17.42668326885173836387493683781, −16.59445880222187204981176218804, −15.38852247067677235876852903302, −14.13774567510940222435000800584, −12.98372377378828430433305407624, −10.73975194143398729373014910770, −10.24206550538813791086446236999, −7.07827110703585845879190658452, −6.12572066303724874117967128787, −4.95605739802800656523572314859, 1.74073379470329693014526709821, 5.06677998530991588708145928072, 6.20514275058079916546051722600, 9.264035863309625285116670997161, 10.83284362973567230368308561097, 11.89678754614274786787956609785, 12.80186223380854215886044059612, 13.66745316229149054327744291367, 16.20808406484904742375210929922, 17.13725019766431333656294214299

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