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Properties

Label 2-575-23.8-c1-0-4
Degree $2$
Conductor $575$
Sign $-0.509 + 0.860i$
Analytic cond. $4.59139$
Root an. cond. $2.14275$
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  + (−0.336 + 2.33i)2-s + (1.10 + 2.42i)3-s + (−3.44 − 1.01i)4-s + (−6.04 + 1.77i)6-s + (2.96 + 1.90i)7-s + (1.55 − 3.41i)8-s + (−2.68 + 3.09i)9-s + (−0.318 − 2.21i)11-s + (−1.36 − 9.46i)12-s + (−2.99 + 1.92i)13-s + (−5.46 + 6.30i)14-s + (1.42 + 0.918i)16-s + (−5.12 + 1.50i)17-s + (−6.33 − 7.31i)18-s + (2.61 + 0.769i)19-s + ⋯
L(s)  = 1  + (−0.237 + 1.65i)2-s + (0.638 + 1.39i)3-s + (−1.72 − 0.505i)4-s + (−2.46 + 0.724i)6-s + (1.12 + 0.720i)7-s + (0.551 − 1.20i)8-s + (−0.893 + 1.03i)9-s + (−0.0959 − 0.667i)11-s + (−0.392 − 2.73i)12-s + (−0.831 + 0.534i)13-s + (−1.45 + 1.68i)14-s + (0.357 + 0.229i)16-s + (−1.24 + 0.365i)17-s + (−1.49 − 1.72i)18-s + (0.600 + 0.176i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $-0.509 + 0.860i$
Analytic conductor: \(4.59139\)
Root analytic conductor: \(2.14275\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{575} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 575,\ (\ :1/2),\ -0.509 + 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.705929 - 1.23798i\)
\(L(\frac12)\) \(\approx\) \(0.705929 - 1.23798i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 + (-1.35 + 4.60i)T \)
good2 \( 1 + (0.336 - 2.33i)T + (-1.91 - 0.563i)T^{2} \)
3 \( 1 + (-1.10 - 2.42i)T + (-1.96 + 2.26i)T^{2} \)
7 \( 1 + (-2.96 - 1.90i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (0.318 + 2.21i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (2.99 - 1.92i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (5.12 - 1.50i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (-2.61 - 0.769i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (-4.25 + 1.24i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (0.941 - 2.06i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (4.24 - 4.90i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (-1.88 - 2.17i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-1.67 - 3.66i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + (-6.11 - 3.93i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-5.02 + 3.23i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (1.60 - 3.51i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (-0.217 + 1.51i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (0.0271 - 0.188i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (4.32 + 1.27i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (4.72 - 3.03i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-4.89 + 5.64i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (2.09 + 4.57i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (6.12 + 7.07i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.02097322730267229939145928465, −10.09372661088366764687487594962, −9.017052983407830183666233604373, −8.732065552352028195010866555619, −8.017786864073962862577908597377, −6.88337326129319088798107098856, −5.72556150827546794029823967012, −4.84477595843234812349365309800, −4.32662969530308997847334965587, −2.61063158080966429485703081425, 0.844233629328679510170402021081, 1.94943137639509041542173193550, 2.64824714742357195903013269595, 4.04390780597287632301825750421, 5.12726717235409873396179478589, 7.07184539053120632269225570988, 7.51154433892875156093543072372, 8.524622310564285895219188577323, 9.310575934314264861718674404403, 10.36200225116828441752330732416

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