Properties

Label 2-23-23.13-c3-0-4
Degree $2$
Conductor $23$
Sign $-0.893 - 0.449i$
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  + (−3.54 − 4.08i)2-s + (−2.50 − 1.60i)3-s + (−3.02 + 21.0i)4-s + (1.32 − 2.90i)5-s + (2.28 + 15.9i)6-s + (−29.0 − 8.52i)7-s + (60.1 − 38.6i)8-s + (−7.53 − 16.5i)9-s + (−16.5 + 4.86i)10-s + (−4.08 + 4.70i)11-s + (41.3 − 47.7i)12-s + (32.2 − 9.47i)13-s + (67.9 + 148. i)14-s + (−8.00 + 5.14i)15-s + (−207. − 61.0i)16-s + (−10.8 − 75.3i)17-s + ⋯
L(s)  = 1  + (−1.25 − 1.44i)2-s + (−0.481 − 0.309i)3-s + (−0.377 + 2.62i)4-s + (0.118 − 0.260i)5-s + (0.155 + 1.08i)6-s + (−1.56 − 0.460i)7-s + (2.65 − 1.70i)8-s + (−0.279 − 0.611i)9-s + (−0.524 + 0.153i)10-s + (−0.111 + 0.129i)11-s + (0.995 − 1.14i)12-s + (0.688 − 0.202i)13-s + (1.29 + 2.84i)14-s + (−0.137 + 0.0885i)15-s + (−3.24 − 0.954i)16-s + (−0.154 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $-0.893 - 0.449i$
Analytic conductor: \(1.35704\)
Root analytic conductor: \(1.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :3/2),\ -0.893 - 0.449i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0711665 + 0.299858i\)
\(L(\frac12)\) \(\approx\) \(0.0711665 + 0.299858i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-50.7 + 97.9i)T \)
good2 \( 1 + (3.54 + 4.08i)T + (-1.13 + 7.91i)T^{2} \)
3 \( 1 + (2.50 + 1.60i)T + (11.2 + 24.5i)T^{2} \)
5 \( 1 + (-1.32 + 2.90i)T + (-81.8 - 94.4i)T^{2} \)
7 \( 1 + (29.0 + 8.52i)T + (288. + 185. i)T^{2} \)
11 \( 1 + (4.08 - 4.70i)T + (-189. - 1.31e3i)T^{2} \)
13 \( 1 + (-32.2 + 9.47i)T + (1.84e3 - 1.18e3i)T^{2} \)
17 \( 1 + (10.8 + 75.3i)T + (-4.71e3 + 1.38e3i)T^{2} \)
19 \( 1 + (1.73 - 12.0i)T + (-6.58e3 - 1.93e3i)T^{2} \)
29 \( 1 + (11.9 + 82.9i)T + (-2.34e4 + 6.87e3i)T^{2} \)
31 \( 1 + (76.3 - 49.0i)T + (1.23e4 - 2.70e4i)T^{2} \)
37 \( 1 + (8.19 + 17.9i)T + (-3.31e4 + 3.82e4i)T^{2} \)
41 \( 1 + (-25.8 + 56.6i)T + (-4.51e4 - 5.20e4i)T^{2} \)
43 \( 1 + (185. + 119. i)T + (3.30e4 + 7.23e4i)T^{2} \)
47 \( 1 + 257.T + 1.03e5T^{2} \)
53 \( 1 + (-8.59 - 2.52i)T + (1.25e5 + 8.04e4i)T^{2} \)
59 \( 1 + (-246. + 72.3i)T + (1.72e5 - 1.11e5i)T^{2} \)
61 \( 1 + (-656. + 422. i)T + (9.42e4 - 2.06e5i)T^{2} \)
67 \( 1 + (283. + 327. i)T + (-4.28e4 + 2.97e5i)T^{2} \)
71 \( 1 + (66.9 + 77.2i)T + (-5.09e4 + 3.54e5i)T^{2} \)
73 \( 1 + (79.9 - 556. i)T + (-3.73e5 - 1.09e5i)T^{2} \)
79 \( 1 + (-782. + 229. i)T + (4.14e5 - 2.66e5i)T^{2} \)
83 \( 1 + (362. + 793. i)T + (-3.74e5 + 4.32e5i)T^{2} \)
89 \( 1 + (524. + 337. i)T + (2.92e5 + 6.41e5i)T^{2} \)
97 \( 1 + (-303. + 663. i)T + (-5.97e5 - 6.89e5i)T^{2} \)
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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

−17.03728319090444485080109038507, −16.18535066246896303677069352830, −13.23318498209750123310705579838, −12.47972103571332875743181914117, −11.20789446729418710445462880904, −9.913791199836414318621579491270, −8.901517660471065024850441421436, −6.90497477272787661613053710829, −3.30077715635471502470867907760, −0.50571586169739045056980740336, 5.68696267127263487190869125837, 6.68945579377188179366448850485, 8.496378860759206584082366312996, 9.760102595491866463057472768858, 10.84944840494114713681894653247, 13.36491150435621631205143549769, 14.97009723073617651065099108595, 16.09240758787087137859854179400, 16.57856227446507314955862357763, 17.83810832496303217319249633241

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