Properties

Label 2-23-23.2-c1-0-0
Degree $2$
Conductor $23$
Sign $0.131 + 0.991i$
Analytic cond. $0.183655$
Root an. cond. $0.428550$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.11 − 1.35i)2-s + (−0.226 − 1.57i)3-s + (1.79 + 3.92i)4-s + (1.41 + 0.416i)5-s + (−1.66 + 3.64i)6-s + (−0.804 + 0.928i)7-s + (0.828 − 5.76i)8-s + (0.439 − 0.129i)9-s + (−2.43 − 2.80i)10-s + (−2.98 + 1.91i)11-s + (5.79 − 3.72i)12-s + (0.0892 + 0.103i)13-s + (2.96 − 0.870i)14-s + (0.335 − 2.33i)15-s + (−3.93 + 4.53i)16-s + (−2.50 + 5.48i)17-s + ⋯
L(s)  = 1  + (−1.49 − 0.960i)2-s + (−0.131 − 0.911i)3-s + (0.896 + 1.96i)4-s + (0.634 + 0.186i)5-s + (−0.679 + 1.48i)6-s + (−0.304 + 0.350i)7-s + (0.292 − 2.03i)8-s + (0.146 − 0.0430i)9-s + (−0.769 − 0.888i)10-s + (−0.899 + 0.578i)11-s + (1.67 − 1.07i)12-s + (0.0247 + 0.0285i)13-s + (0.791 − 0.232i)14-s + (0.0866 − 0.602i)15-s + (−0.982 + 1.13i)16-s + (−0.607 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.131 + 0.991i$
Analytic conductor: \(0.183655\)
Root analytic conductor: \(0.428550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :1/2),\ 0.131 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.273222 - 0.239463i\)
\(L(\frac12)\) \(\approx\) \(0.273222 - 0.239463i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-4.66 + 1.11i)T \)
good2 \( 1 + (2.11 + 1.35i)T + (0.830 + 1.81i)T^{2} \)
3 \( 1 + (0.226 + 1.57i)T + (-2.87 + 0.845i)T^{2} \)
5 \( 1 + (-1.41 - 0.416i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (0.804 - 0.928i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (2.98 - 1.91i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (-0.0892 - 0.103i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (2.50 - 5.48i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (1.33 + 2.92i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (-1.90 + 4.18i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.409 - 2.84i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (3.73 - 1.09i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (0.981 + 0.288i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-0.450 - 3.13i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 - 0.00935T + 47T^{2} \)
53 \( 1 + (-6.16 + 7.11i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (3.05 + 3.52i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (-1.92 + 13.4i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (-8.18 - 5.25i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (0.585 + 0.376i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (2.20 + 4.82i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (-7.03 - 8.11i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (-1.68 + 0.493i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-1.25 - 8.72i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (17.2 + 5.07i)T + (81.6 + 52.4i)T^{2} \)
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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

−17.86147880639959092364631868931, −17.31099181088647108588348583878, −15.55106807150747129847121345032, −13.17407719186635199298694139856, −12.37476883078098682361873215131, −10.81810429255916302853947500307, −9.705579411368379280729362111163, −8.230862233098849007811223164072, −6.72834303194291122857700768805, −2.19372870353046740349042896222, 5.43877314781797226185023792376, 7.21201196922533576535937855384, 8.911451895775021799548826470980, 9.940318223590862019086031821196, 10.83003785505198518881029251456, 13.52192718420478674356555025850, 15.21286790608902021885181757823, 16.13006173285204030839300046920, 16.81455598106203591199972402870, 18.02465218050932351203487499531

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