Properties

Label 2-23-23.19-c2-0-1
Degree $2$
Conductor $23$
Sign $0.916 + 0.399i$
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  + (−0.282 − 0.618i)2-s + (0.844 − 0.247i)3-s + (2.31 − 2.67i)4-s + (−3.24 + 5.05i)5-s + (−0.391 − 0.452i)6-s + (−3.20 + 0.460i)7-s + (−4.91 − 1.44i)8-s + (−6.91 + 4.44i)9-s + (4.04 + 0.580i)10-s + (10.2 + 4.66i)11-s + (1.29 − 2.83i)12-s + (2.01 − 13.9i)13-s + (1.18 + 1.84i)14-s + (−1.48 + 5.07i)15-s + (−1.51 − 10.5i)16-s + (9.44 − 8.18i)17-s + ⋯
L(s)  = 1  + (−0.141 − 0.309i)2-s + (0.281 − 0.0826i)3-s + (0.579 − 0.668i)4-s + (−0.649 + 1.01i)5-s + (−0.0652 − 0.0753i)6-s + (−0.457 + 0.0657i)7-s + (−0.614 − 0.180i)8-s + (−0.768 + 0.494i)9-s + (0.404 + 0.0580i)10-s + (0.928 + 0.424i)11-s + (0.107 − 0.236i)12-s + (0.154 − 1.07i)13-s + (0.0848 + 0.132i)14-s + (−0.0993 + 0.338i)15-s + (−0.0949 − 0.660i)16-s + (0.555 − 0.481i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23\)
Sign: $0.916 + 0.399i$
Analytic conductor: \(0.626704\)
Root analytic conductor: \(0.791646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 23,\ (\ :1),\ 0.916 + 0.399i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.876852 - 0.183001i\)
\(L(\frac12)\) \(\approx\) \(0.876852 - 0.183001i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (22.9 - 0.0832i)T \)
good2 \( 1 + (0.282 + 0.618i)T + (-2.61 + 3.02i)T^{2} \)
3 \( 1 + (-0.844 + 0.247i)T + (7.57 - 4.86i)T^{2} \)
5 \( 1 + (3.24 - 5.05i)T + (-10.3 - 22.7i)T^{2} \)
7 \( 1 + (3.20 - 0.460i)T + (47.0 - 13.8i)T^{2} \)
11 \( 1 + (-10.2 - 4.66i)T + (79.2 + 91.4i)T^{2} \)
13 \( 1 + (-2.01 + 13.9i)T + (-162. - 47.6i)T^{2} \)
17 \( 1 + (-9.44 + 8.18i)T + (41.1 - 286. i)T^{2} \)
19 \( 1 + (-5.84 - 5.06i)T + (51.3 + 357. i)T^{2} \)
29 \( 1 + (20.6 + 23.8i)T + (-119. + 832. i)T^{2} \)
31 \( 1 + (-58.8 - 17.2i)T + (808. + 519. i)T^{2} \)
37 \( 1 + (26.1 + 40.6i)T + (-568. + 1.24e3i)T^{2} \)
41 \( 1 + (-38.1 - 24.5i)T + (698. + 1.52e3i)T^{2} \)
43 \( 1 + (-16.7 - 56.9i)T + (-1.55e3 + 9.99e2i)T^{2} \)
47 \( 1 - 3.02T + 2.20e3T^{2} \)
53 \( 1 + (41.9 - 6.03i)T + (2.69e3 - 791. i)T^{2} \)
59 \( 1 + (-4.28 + 29.7i)T + (-3.33e3 - 980. i)T^{2} \)
61 \( 1 + (26.7 - 91.2i)T + (-3.13e3 - 2.01e3i)T^{2} \)
67 \( 1 + (30.1 - 13.7i)T + (2.93e3 - 3.39e3i)T^{2} \)
71 \( 1 + (10.9 + 23.9i)T + (-3.30e3 + 3.80e3i)T^{2} \)
73 \( 1 + (-53.2 + 61.4i)T + (-758. - 5.27e3i)T^{2} \)
79 \( 1 + (-86.0 - 12.3i)T + (5.98e3 + 1.75e3i)T^{2} \)
83 \( 1 + (0.815 + 1.26i)T + (-2.86e3 + 6.26e3i)T^{2} \)
89 \( 1 + (11.3 + 38.5i)T + (-6.66e3 + 4.28e3i)T^{2} \)
97 \( 1 + (15.4 - 24.1i)T + (-3.90e3 - 8.55e3i)T^{2} \)
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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

−17.86469621759433652474535766626, −16.09917017402544923731311067653, −14.99829282863003277446233461747, −14.10972244247811517216449709807, −12.04954272822936003656974322604, −10.99217932437499163258735669062, −9.752283209870109989014742150249, −7.68979260404592507820928612840, −6.11913210485794748419011819568, −3.00937815126883316297435672645, 3.73974811544193293319662335732, 6.39940565614267421930043451499, 8.193417611882323813239580580643, 9.152961762094982066622187770183, 11.61966839705951180769401147778, 12.29738789896160647827677217461, 14.05327718358201263290720254826, 15.57880250805784303491238567009, 16.54673739648348204414918905427, 17.24857073612493575930589735404

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