Properties

Label 2-23-23.14-c2-0-0
Degree $2$
Conductor $23$
Sign $0.622 - 0.782i$
Analytic cond. $0.626704$
Root an. cond. $0.791646$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 + 0.528i)2-s + (2.35 + 2.72i)3-s + (−0.402 + 0.258i)4-s + (5.05 − 0.726i)5-s + (−5.68 − 3.65i)6-s + (−8.85 − 4.04i)7-s + (5.50 − 6.35i)8-s + (−0.563 + 3.92i)9-s + (−8.71 + 3.97i)10-s + (2.47 − 8.43i)11-s + (−1.65 − 0.485i)12-s + (5.88 + 12.8i)13-s + (18.0 + 2.59i)14-s + (13.8 + 12.0i)15-s + (−5.75 + 12.6i)16-s + (−3.89 + 6.05i)17-s + ⋯
L(s)  = 1  + (−0.900 + 0.264i)2-s + (0.785 + 0.906i)3-s + (−0.100 + 0.0646i)4-s + (1.01 − 0.145i)5-s + (−0.947 − 0.608i)6-s + (−1.26 − 0.577i)7-s + (0.687 − 0.793i)8-s + (−0.0626 + 0.435i)9-s + (−0.871 + 0.397i)10-s + (0.225 − 0.766i)11-s + (−0.137 − 0.0404i)12-s + (0.453 + 0.992i)13-s + (1.29 + 0.185i)14-s + (0.925 + 0.802i)15-s + (−0.359 + 0.787i)16-s + (−0.228 + 0.356i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.622 - 0.782i$
Analytic conductor: \(0.626704\)
Root analytic conductor: \(0.791646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :1),\ 0.622 - 0.782i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.655999 + 0.316430i\)
\(L(\frac12)\) \(\approx\) \(0.655999 + 0.316430i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (11.2 - 20.0i)T \)
good2 \( 1 + (1.80 - 0.528i)T + (3.36 - 2.16i)T^{2} \)
3 \( 1 + (-2.35 - 2.72i)T + (-1.28 + 8.90i)T^{2} \)
5 \( 1 + (-5.05 + 0.726i)T + (23.9 - 7.04i)T^{2} \)
7 \( 1 + (8.85 + 4.04i)T + (32.0 + 37.0i)T^{2} \)
11 \( 1 + (-2.47 + 8.43i)T + (-101. - 65.4i)T^{2} \)
13 \( 1 + (-5.88 - 12.8i)T + (-110. + 127. i)T^{2} \)
17 \( 1 + (3.89 - 6.05i)T + (-120. - 262. i)T^{2} \)
19 \( 1 + (15.1 + 23.5i)T + (-149. + 328. i)T^{2} \)
29 \( 1 + (-10.1 - 6.53i)T + (349. + 765. i)T^{2} \)
31 \( 1 + (28.5 - 33.0i)T + (-136. - 951. i)T^{2} \)
37 \( 1 + (-32.2 - 4.64i)T + (1.31e3 + 385. i)T^{2} \)
41 \( 1 + (8.24 + 57.3i)T + (-1.61e3 + 473. i)T^{2} \)
43 \( 1 + (6.69 - 5.80i)T + (263. - 1.83e3i)T^{2} \)
47 \( 1 - 10.4T + 2.20e3T^{2} \)
53 \( 1 + (-20.7 - 9.47i)T + (1.83e3 + 2.12e3i)T^{2} \)
59 \( 1 + (3.35 + 7.34i)T + (-2.27e3 + 2.63e3i)T^{2} \)
61 \( 1 + (13.7 + 11.9i)T + (529. + 3.68e3i)T^{2} \)
67 \( 1 + (-21.0 - 71.8i)T + (-3.77e3 + 2.42e3i)T^{2} \)
71 \( 1 + (-48.3 + 14.2i)T + (4.24e3 - 2.72e3i)T^{2} \)
73 \( 1 + (61.7 - 39.6i)T + (2.21e3 - 4.84e3i)T^{2} \)
79 \( 1 + (-56.1 + 25.6i)T + (4.08e3 - 4.71e3i)T^{2} \)
83 \( 1 + (-75.3 - 10.8i)T + (6.60e3 + 1.94e3i)T^{2} \)
89 \( 1 + (-44.2 + 38.3i)T + (1.12e3 - 7.84e3i)T^{2} \)
97 \( 1 + (-8.57 + 1.23i)T + (9.02e3 - 2.65e3i)T^{2} \)
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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

−17.64501473586825050572171577547, −16.59786811351257284805523999797, −15.79289143975743091969442792753, −13.96124861552244661117672895214, −13.18600554104263312729365745539, −10.48153350953569057484091636453, −9.398992014766000359357446323822, −8.861834134248371969893479674829, −6.63036115670050442036208586432, −3.82398591001900582092916963114, 2.23517194068558432823401597728, 6.16852793794585659097726268041, 8.038592810577284235958890652924, 9.360201152705622333750371558854, 10.26951139093920988120456206982, 12.65476840406499913034258860338, 13.49444329547319297878175829687, 14.74860448443699443343844013003, 16.63802068256688144126378529306, 18.07119280168235497521380813202

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