Properties

Label 2-23-23.18-c3-0-3
Degree $2$
Conductor $23$
Sign $-0.00677 + 0.999i$
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  + (−2.49 − 0.733i)2-s + (2.24 − 2.59i)3-s + (−1.02 − 0.661i)4-s + (2.23 − 15.5i)5-s + (−7.50 + 4.82i)6-s + (−2.46 − 5.40i)7-s + (15.7 + 18.1i)8-s + (2.16 + 15.0i)9-s + (−16.9 + 37.1i)10-s + (20.9 − 6.16i)11-s + (−4.02 + 1.18i)12-s + (−14.2 + 31.2i)13-s + (2.20 + 15.3i)14-s + (−35.2 − 40.6i)15-s + (−21.9 − 47.9i)16-s + (108. − 69.9i)17-s + ⋯
L(s)  = 1  + (−0.883 − 0.259i)2-s + (0.432 − 0.498i)3-s + (−0.128 − 0.0826i)4-s + (0.199 − 1.38i)5-s + (−0.510 + 0.328i)6-s + (−0.133 − 0.292i)7-s + (0.694 + 0.801i)8-s + (0.0803 + 0.558i)9-s + (−0.536 + 1.17i)10-s + (0.575 − 0.168i)11-s + (−0.0967 + 0.0284i)12-s + (−0.304 + 0.666i)13-s + (0.0420 + 0.292i)14-s + (−0.606 − 0.699i)15-s + (−0.342 − 0.749i)16-s + (1.55 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00677 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00677 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.00677 + 0.999i$
Analytic conductor: \(1.35704\)
Root analytic conductor: \(1.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :3/2),\ -0.00677 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.561531 - 0.565347i\)
\(L(\frac12)\) \(\approx\) \(0.561531 - 0.565347i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (18.9 - 108. i)T \)
good2 \( 1 + (2.49 + 0.733i)T + (6.73 + 4.32i)T^{2} \)
3 \( 1 + (-2.24 + 2.59i)T + (-3.84 - 26.7i)T^{2} \)
5 \( 1 + (-2.23 + 15.5i)T + (-119. - 35.2i)T^{2} \)
7 \( 1 + (2.46 + 5.40i)T + (-224. + 259. i)T^{2} \)
11 \( 1 + (-20.9 + 6.16i)T + (1.11e3 - 719. i)T^{2} \)
13 \( 1 + (14.2 - 31.2i)T + (-1.43e3 - 1.66e3i)T^{2} \)
17 \( 1 + (-108. + 69.9i)T + (2.04e3 - 4.46e3i)T^{2} \)
19 \( 1 + (23.4 + 15.0i)T + (2.84e3 + 6.23e3i)T^{2} \)
29 \( 1 + (-10.5 + 6.78i)T + (1.01e4 - 2.21e4i)T^{2} \)
31 \( 1 + (10.3 + 11.9i)T + (-4.23e3 + 2.94e4i)T^{2} \)
37 \( 1 + (-33.6 - 233. i)T + (-4.86e4 + 1.42e4i)T^{2} \)
41 \( 1 + (-53.6 + 373. i)T + (-6.61e4 - 1.94e4i)T^{2} \)
43 \( 1 + (310. - 358. i)T + (-1.13e4 - 7.86e4i)T^{2} \)
47 \( 1 + 547.T + 1.03e5T^{2} \)
53 \( 1 + (-192. - 421. i)T + (-9.74e4 + 1.12e5i)T^{2} \)
59 \( 1 + (248. - 543. i)T + (-1.34e5 - 1.55e5i)T^{2} \)
61 \( 1 + (181. + 209. i)T + (-3.23e4 + 2.24e5i)T^{2} \)
67 \( 1 + (-164. - 48.3i)T + (2.53e5 + 1.62e5i)T^{2} \)
71 \( 1 + (218. + 64.1i)T + (3.01e5 + 1.93e5i)T^{2} \)
73 \( 1 + (-784. - 504. i)T + (1.61e5 + 3.53e5i)T^{2} \)
79 \( 1 + (-97.3 + 213. i)T + (-3.22e5 - 3.72e5i)T^{2} \)
83 \( 1 + (129. + 903. i)T + (-5.48e5 + 1.61e5i)T^{2} \)
89 \( 1 + (-451. + 521. i)T + (-1.00e5 - 6.97e5i)T^{2} \)
97 \( 1 + (12.2 - 85.2i)T + (-8.75e5 - 2.57e5i)T^{2} \)
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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

−17.00243112452773021926783368723, −16.46905661680525233262427177840, −14.17029627183193125450127409882, −13.31654088458224541282115547870, −11.80611610011734473275407207011, −9.891542764499295831029153282097, −8.887979849604191670239962695282, −7.68288094246071869841921418617, −4.99694317485523673094832096450, −1.33630540378028696921990689848, 3.51525447791054427081600141229, 6.56244960312268133633937548461, 8.144191980295399108924827557687, 9.662683917208372465595489552313, 10.43946876950410506771289824252, 12.49249904052333552243222524969, 14.40275838931986332200668377872, 15.07707276108789923830943858952, 16.63261149224865678956677936583, 17.86152836375530203688813385156

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