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Properties

Label 2-875-175.117-c1-0-29
Degree $2$
Conductor $875$
Sign $0.520 - 0.854i$
Analytic cond. $6.98691$
Root an. cond. $2.64327$
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  + (0.973 + 1.49i)2-s + (0.128 + 0.335i)3-s + (−0.487 + 1.09i)4-s + (−0.378 + 0.520i)6-s + (0.314 − 2.62i)7-s + (1.41 − 0.224i)8-s + (2.13 − 1.92i)9-s + (−2.82 + 3.13i)11-s + (−0.430 − 0.0225i)12-s + (0.527 − 0.268i)13-s + (4.24 − 2.08i)14-s + (3.31 + 3.68i)16-s + (0.546 + 0.675i)17-s + (4.95 + 1.32i)18-s + (5.83 − 2.59i)19-s + ⋯
L(s)  = 1  + (0.688 + 1.06i)2-s + (0.0744 + 0.193i)3-s + (−0.243 + 0.547i)4-s + (−0.154 + 0.212i)6-s + (0.119 − 0.992i)7-s + (0.500 − 0.0793i)8-s + (0.711 − 0.640i)9-s + (−0.851 + 0.946i)11-s + (−0.124 − 0.00651i)12-s + (0.146 − 0.0745i)13-s + (1.13 − 0.557i)14-s + (0.829 + 0.921i)16-s + (0.132 + 0.163i)17-s + (1.16 + 0.313i)18-s + (1.33 − 0.595i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(875\)    =    \(5^{3} \cdot 7\)
Sign: $0.520 - 0.854i$
Analytic conductor: \(6.98691\)
Root analytic conductor: \(2.64327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{875} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 875,\ (\ :1/2),\ 0.520 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28077 + 1.28160i\)
\(L(\frac12)\) \(\approx\) \(2.28077 + 1.28160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (-0.314 + 2.62i)T \)
good2 \( 1 + (-0.973 - 1.49i)T + (-0.813 + 1.82i)T^{2} \)
3 \( 1 + (-0.128 - 0.335i)T + (-2.22 + 2.00i)T^{2} \)
11 \( 1 + (2.82 - 3.13i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (-0.527 + 0.268i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.546 - 0.675i)T + (-3.53 + 16.6i)T^{2} \)
19 \( 1 + (-5.83 + 2.59i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (1.35 - 0.882i)T + (9.35 - 21.0i)T^{2} \)
29 \( 1 + (-0.996 - 1.37i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-8.94 - 0.940i)T + (30.3 + 6.44i)T^{2} \)
37 \( 1 + (-0.228 + 4.35i)T + (-36.7 - 3.86i)T^{2} \)
41 \( 1 + (-5.70 - 1.85i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (3.73 - 3.73i)T - 43iT^{2} \)
47 \( 1 + (7.49 + 6.06i)T + (9.77 + 45.9i)T^{2} \)
53 \( 1 + (7.91 - 3.03i)T + (39.3 - 35.4i)T^{2} \)
59 \( 1 + (6.62 - 1.40i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (-0.256 + 1.20i)T + (-55.7 - 24.8i)T^{2} \)
67 \( 1 + (8.90 - 7.21i)T + (13.9 - 65.5i)T^{2} \)
71 \( 1 + (11.3 - 8.23i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.48 + 0.182i)T + (72.6 - 7.63i)T^{2} \)
79 \( 1 + (-14.0 + 1.47i)T + (77.2 - 16.4i)T^{2} \)
83 \( 1 + (0.477 + 3.01i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (16.6 + 3.54i)T + (81.3 + 36.1i)T^{2} \)
97 \( 1 + (-1.83 + 11.5i)T + (-92.2 - 29.9i)T^{2} \)
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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

−10.09509734885305693232152122341, −9.670846105206834868501859117479, −8.200165729004849967384779321844, −7.40219909865939286636902139907, −6.97233123198647789035557539990, −5.98422361959103580259962776712, −4.83026174093674720634707936446, −4.40331560411110195047119730174, −3.22373027048558454886008261142, −1.30701485940172973084079097406, 1.41955802801348511728157184706, 2.58727795826570099055049287740, 3.27645602604473760000625012686, 4.62165373736212764281141980844, 5.30012497219942420296272134343, 6.31072922267606591118660933236, 7.83118256346169128418239072899, 8.067588189023610569909105010530, 9.452766447676981419805827167226, 10.24913546725306932661011068805

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