Properties

Label 2-23-23.8-c3-0-4
Degree $2$
Conductor $23$
Sign $-0.293 + 0.956i$
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  + (0.619 − 4.31i)2-s + (−0.504 − 1.10i)3-s + (−10.5 − 3.09i)4-s + (4.52 + 5.22i)5-s + (−5.07 + 1.49i)6-s + (2.36 + 1.52i)7-s + (−5.38 + 11.7i)8-s + (16.7 − 19.2i)9-s + (25.3 − 16.2i)10-s + (8.72 + 60.6i)11-s + (1.89 + 13.1i)12-s + (−8.84 + 5.68i)13-s + (8.03 − 9.26i)14-s + (3.48 − 7.63i)15-s + (−26.3 − 16.9i)16-s + (−99.9 + 29.3i)17-s + ⋯
L(s)  = 1  + (0.219 − 1.52i)2-s + (−0.0971 − 0.212i)3-s + (−1.31 − 0.386i)4-s + (0.404 + 0.466i)5-s + (−0.345 + 0.101i)6-s + (0.127 + 0.0821i)7-s + (−0.237 + 0.520i)8-s + (0.619 − 0.714i)9-s + (0.800 − 0.514i)10-s + (0.239 + 1.66i)11-s + (0.0456 + 0.317i)12-s + (−0.188 + 0.121i)13-s + (0.153 − 0.176i)14-s + (0.0600 − 0.131i)15-s + (−0.412 − 0.264i)16-s + (−1.42 + 0.418i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.293 + 0.956i$
Analytic conductor: \(1.35704\)
Root analytic conductor: \(1.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :3/2),\ -0.293 + 0.956i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.738000 - 0.998078i\)
\(L(\frac12)\) \(\approx\) \(0.738000 - 0.998078i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (109. - 10.3i)T \)
good2 \( 1 + (-0.619 + 4.31i)T + (-7.67 - 2.25i)T^{2} \)
3 \( 1 + (0.504 + 1.10i)T + (-17.6 + 20.4i)T^{2} \)
5 \( 1 + (-4.52 - 5.22i)T + (-17.7 + 123. i)T^{2} \)
7 \( 1 + (-2.36 - 1.52i)T + (142. + 312. i)T^{2} \)
11 \( 1 + (-8.72 - 60.6i)T + (-1.27e3 + 374. i)T^{2} \)
13 \( 1 + (8.84 - 5.68i)T + (912. - 1.99e3i)T^{2} \)
17 \( 1 + (99.9 - 29.3i)T + (4.13e3 - 2.65e3i)T^{2} \)
19 \( 1 + (-67.6 - 19.8i)T + (5.77e3 + 3.70e3i)T^{2} \)
29 \( 1 + (-118. + 34.9i)T + (2.05e4 - 1.31e4i)T^{2} \)
31 \( 1 + (60.1 - 131. i)T + (-1.95e4 - 2.25e4i)T^{2} \)
37 \( 1 + (-150. + 174. i)T + (-7.20e3 - 5.01e4i)T^{2} \)
41 \( 1 + (13.1 + 15.2i)T + (-9.80e3 + 6.82e4i)T^{2} \)
43 \( 1 + (92.1 + 201. i)T + (-5.20e4 + 6.00e4i)T^{2} \)
47 \( 1 - 487.T + 1.03e5T^{2} \)
53 \( 1 + (317. + 204. i)T + (6.18e4 + 1.35e5i)T^{2} \)
59 \( 1 + (561. - 361. i)T + (8.53e4 - 1.86e5i)T^{2} \)
61 \( 1 + (-31.0 + 67.9i)T + (-1.48e5 - 1.71e5i)T^{2} \)
67 \( 1 + (-9.35 + 65.0i)T + (-2.88e5 - 8.47e4i)T^{2} \)
71 \( 1 + (93.1 - 648. i)T + (-3.43e5 - 1.00e5i)T^{2} \)
73 \( 1 + (-578. - 169. i)T + (3.27e5 + 2.10e5i)T^{2} \)
79 \( 1 + (-98.2 + 63.1i)T + (2.04e5 - 4.48e5i)T^{2} \)
83 \( 1 + (203. - 235. i)T + (-8.13e4 - 5.65e5i)T^{2} \)
89 \( 1 + (-631. - 1.38e3i)T + (-4.61e5 + 5.32e5i)T^{2} \)
97 \( 1 + (545. + 629. i)T + (-1.29e5 + 9.03e5i)T^{2} \)
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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

−17.80417713641383568122761663126, −15.50949556246079376919526235066, −14.08136265078636505713452497906, −12.69812971574042464411934836314, −11.92110480113972674165670235911, −10.39381403717361513178706878393, −9.472647377366289439169644409234, −6.87269555246052917108385290914, −4.25901785424470624081140913421, −2.02624449627305789770829756740, 4.77157242548140867271413280309, 6.14739583293634228559713806790, 7.80632577337575738237082888230, 9.144911393565947554201420791956, 11.11459200351095101061734062007, 13.33684920942174917807206056791, 13.97284700227886685989771682982, 15.61144085236122404329315203439, 16.31728048030420354614763416531, 17.24429768941653561739004071589

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