Properties

Label 2-23-23.3-c3-0-0
Degree $2$
Conductor $23$
Sign $0.308 - 0.951i$
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  + (0.0282 + 0.196i)2-s + (−3.66 + 8.01i)3-s + (7.63 − 2.24i)4-s + (−6.59 + 7.60i)5-s + (−1.67 − 0.493i)6-s + (21.3 − 13.7i)7-s + (1.31 + 2.88i)8-s + (−33.1 − 38.2i)9-s + (−1.68 − 1.08i)10-s + (2.01 − 14.0i)11-s + (−9.98 + 69.4i)12-s + (−3.85 − 2.47i)13-s + (3.29 + 3.80i)14-s + (−36.8 − 80.6i)15-s + (53.0 − 34.0i)16-s + (98.0 + 28.7i)17-s + ⋯
L(s)  = 1  + (0.00999 + 0.0695i)2-s + (−0.704 + 1.54i)3-s + (0.954 − 0.280i)4-s + (−0.589 + 0.680i)5-s + (−0.114 − 0.0335i)6-s + (1.15 − 0.739i)7-s + (0.0582 + 0.127i)8-s + (−1.22 − 1.41i)9-s + (−0.0532 − 0.0341i)10-s + (0.0552 − 0.384i)11-s + (−0.240 + 1.67i)12-s + (−0.0822 − 0.0528i)13-s + (0.0629 + 0.0726i)14-s + (−0.634 − 1.38i)15-s + (0.828 − 0.532i)16-s + (1.39 + 0.410i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.308 - 0.951i$
Analytic conductor: \(1.35704\)
Root analytic conductor: \(1.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :3/2),\ 0.308 - 0.951i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.876567 + 0.637009i\)
\(L(\frac12)\) \(\approx\) \(0.876567 + 0.637009i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (83.9 + 71.5i)T \)
good2 \( 1 + (-0.0282 - 0.196i)T + (-7.67 + 2.25i)T^{2} \)
3 \( 1 + (3.66 - 8.01i)T + (-17.6 - 20.4i)T^{2} \)
5 \( 1 + (6.59 - 7.60i)T + (-17.7 - 123. i)T^{2} \)
7 \( 1 + (-21.3 + 13.7i)T + (142. - 312. i)T^{2} \)
11 \( 1 + (-2.01 + 14.0i)T + (-1.27e3 - 374. i)T^{2} \)
13 \( 1 + (3.85 + 2.47i)T + (912. + 1.99e3i)T^{2} \)
17 \( 1 + (-98.0 - 28.7i)T + (4.13e3 + 2.65e3i)T^{2} \)
19 \( 1 + (66.4 - 19.5i)T + (5.77e3 - 3.70e3i)T^{2} \)
29 \( 1 + (202. + 59.4i)T + (2.05e4 + 1.31e4i)T^{2} \)
31 \( 1 + (117. + 256. i)T + (-1.95e4 + 2.25e4i)T^{2} \)
37 \( 1 + (-114. - 131. i)T + (-7.20e3 + 5.01e4i)T^{2} \)
41 \( 1 + (66.9 - 77.2i)T + (-9.80e3 - 6.82e4i)T^{2} \)
43 \( 1 + (-53.4 + 116. i)T + (-5.20e4 - 6.00e4i)T^{2} \)
47 \( 1 - 130.T + 1.03e5T^{2} \)
53 \( 1 + (81.5 - 52.3i)T + (6.18e4 - 1.35e5i)T^{2} \)
59 \( 1 + (-232. - 149. i)T + (8.53e4 + 1.86e5i)T^{2} \)
61 \( 1 + (-190. - 416. i)T + (-1.48e5 + 1.71e5i)T^{2} \)
67 \( 1 + (-49.8 - 346. i)T + (-2.88e5 + 8.47e4i)T^{2} \)
71 \( 1 + (109. + 758. i)T + (-3.43e5 + 1.00e5i)T^{2} \)
73 \( 1 + (869. - 255. i)T + (3.27e5 - 2.10e5i)T^{2} \)
79 \( 1 + (-377. - 242. i)T + (2.04e5 + 4.48e5i)T^{2} \)
83 \( 1 + (281. + 325. i)T + (-8.13e4 + 5.65e5i)T^{2} \)
89 \( 1 + (138. - 304. i)T + (-4.61e5 - 5.32e5i)T^{2} \)
97 \( 1 + (557. - 643. i)T + (-1.29e5 - 9.03e5i)T^{2} \)
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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

−17.04254317661647607923942580411, −16.45100993370595708075756729466, −15.01477431204622249348586095311, −14.67039533803447125085465009733, −11.67141840969087897045557356030, −10.96485744848147385369296164470, −10.16281833463862138104411933134, −7.73291008711341703384675052604, −5.76861665654945848722486832347, −3.93850361285048538275506321435, 1.73584609366888119606850895528, 5.55177791440571751632964091513, 7.28361987943604452994669976214, 8.209788716578022339569693349357, 11.20004020622204992879477729570, 12.06898481777303903534359631489, 12.62114664500854587978851585014, 14.57264011556464627456723039742, 16.15237705714742371341247459257, 17.26056155001359908029069293332

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