Properties

Label 2-23-23.16-c3-0-4
Degree $2$
Conductor $23$
Sign $0.300 + 0.953i$
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  + (3.50 − 4.04i)2-s + (−4.18 + 2.69i)3-s + (−2.93 − 20.3i)4-s + (6.92 + 15.1i)5-s + (−3.78 + 26.3i)6-s + (7.38 − 2.16i)7-s + (−56.7 − 36.4i)8-s + (−0.923 + 2.02i)9-s + (85.5 + 25.1i)10-s + (−12.2 − 14.0i)11-s + (67.1 + 77.5i)12-s + (−28.3 − 8.33i)13-s + (17.0 − 37.4i)14-s + (−69.8 − 44.8i)15-s + (−187. + 55.1i)16-s + (−6.29 + 43.7i)17-s + ⋯
L(s)  = 1  + (1.23 − 1.42i)2-s + (−0.805 + 0.517i)3-s + (−0.366 − 2.54i)4-s + (0.619 + 1.35i)5-s + (−0.257 + 1.79i)6-s + (0.398 − 0.117i)7-s + (−2.50 − 1.61i)8-s + (−0.0342 + 0.0748i)9-s + (2.70 + 0.794i)10-s + (−0.334 − 0.386i)11-s + (1.61 + 1.86i)12-s + (−0.605 − 0.177i)13-s + (0.326 − 0.714i)14-s + (−1.20 − 0.772i)15-s + (−2.93 + 0.861i)16-s + (−0.0898 + 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.300 + 0.953i$
Analytic conductor: \(1.35704\)
Root analytic conductor: \(1.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :3/2),\ 0.300 + 0.953i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.30572 - 0.957279i\)
\(L(\frac12)\) \(\approx\) \(1.30572 - 0.957279i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-104. + 34.4i)T \)
good2 \( 1 + (-3.50 + 4.04i)T + (-1.13 - 7.91i)T^{2} \)
3 \( 1 + (4.18 - 2.69i)T + (11.2 - 24.5i)T^{2} \)
5 \( 1 + (-6.92 - 15.1i)T + (-81.8 + 94.4i)T^{2} \)
7 \( 1 + (-7.38 + 2.16i)T + (288. - 185. i)T^{2} \)
11 \( 1 + (12.2 + 14.0i)T + (-189. + 1.31e3i)T^{2} \)
13 \( 1 + (28.3 + 8.33i)T + (1.84e3 + 1.18e3i)T^{2} \)
17 \( 1 + (6.29 - 43.7i)T + (-4.71e3 - 1.38e3i)T^{2} \)
19 \( 1 + (8.21 + 57.1i)T + (-6.58e3 + 1.93e3i)T^{2} \)
29 \( 1 + (-18.6 + 129. i)T + (-2.34e4 - 6.87e3i)T^{2} \)
31 \( 1 + (-128. - 82.3i)T + (1.23e4 + 2.70e4i)T^{2} \)
37 \( 1 + (35.6 - 78.0i)T + (-3.31e4 - 3.82e4i)T^{2} \)
41 \( 1 + (-138. - 304. i)T + (-4.51e4 + 5.20e4i)T^{2} \)
43 \( 1 + (168. - 108. i)T + (3.30e4 - 7.23e4i)T^{2} \)
47 \( 1 - 126.T + 1.03e5T^{2} \)
53 \( 1 + (-728. + 213. i)T + (1.25e5 - 8.04e4i)T^{2} \)
59 \( 1 + (711. + 208. i)T + (1.72e5 + 1.11e5i)T^{2} \)
61 \( 1 + (221. + 142. i)T + (9.42e4 + 2.06e5i)T^{2} \)
67 \( 1 + (-66.9 + 77.2i)T + (-4.28e4 - 2.97e5i)T^{2} \)
71 \( 1 + (180. - 208. i)T + (-5.09e4 - 3.54e5i)T^{2} \)
73 \( 1 + (-86.8 - 604. i)T + (-3.73e5 + 1.09e5i)T^{2} \)
79 \( 1 + (954. + 280. i)T + (4.14e5 + 2.66e5i)T^{2} \)
83 \( 1 + (-154. + 337. i)T + (-3.74e5 - 4.32e5i)T^{2} \)
89 \( 1 + (-630. + 404. i)T + (2.92e5 - 6.41e5i)T^{2} \)
97 \( 1 + (153. + 337. i)T + (-5.97e5 + 6.89e5i)T^{2} \)
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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

−17.40368678889550715077284197638, −15.28260594643182834865367188626, −14.31469679345750467250407657695, −13.23280933961364001035476405716, −11.56704313918216150222393406290, −10.77968735149823655078298974530, −10.07043371694425682489871986226, −6.20195830813757896180853743769, −4.82505251734873678412291096532, −2.72713363669187928373647225172, 4.84447338080152930440454015934, 5.72065386218414150525307691457, 7.25997703259715352777940620042, 8.891703527710865757662586729469, 12.00906610424892587299200246221, 12.68305953965038576644477047700, 13.76587191339510110715519441027, 15.12941844868796853326315484818, 16.52536951705759889638632310888, 17.13760284763005982427172328043

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