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Properties

Label 2-875-175.152-c1-0-20
Degree $2$
Conductor $875$
Sign $-0.297 - 0.954i$
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  + (0.00351 − 0.00434i)2-s + (1.74 + 1.13i)3-s + (0.415 + 1.95i)4-s + (0.0110 − 0.00359i)6-s + (1.62 + 2.08i)7-s + (0.0199 + 0.0101i)8-s + (0.540 + 1.21i)9-s + (−3.82 − 1.70i)11-s + (−1.49 + 3.88i)12-s + (0.121 + 0.769i)13-s + (0.0147 + 0.000255i)14-s + (−3.65 + 1.62i)16-s + (0.213 + 4.07i)17-s + (0.00717 + 0.00192i)18-s + (3.90 + 0.830i)19-s + ⋯
L(s)  = 1  + (0.00248 − 0.00307i)2-s + (1.00 + 0.654i)3-s + (0.207 + 0.978i)4-s + (0.00451 − 0.00146i)6-s + (0.615 + 0.787i)7-s + (0.00703 + 0.00358i)8-s + (0.180 + 0.404i)9-s + (−1.15 − 0.513i)11-s + (−0.430 + 1.12i)12-s + (0.0338 + 0.213i)13-s + (0.00394 + 6.81e−5i)14-s + (−0.913 + 0.406i)16-s + (0.0517 + 0.987i)17-s + (0.00169 + 0.000453i)18-s + (0.895 + 0.190i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(875\)    =    \(5^{3} \cdot 7\)
Sign: $-0.297 - 0.954i$
Analytic conductor: \(6.98691\)
Root analytic conductor: \(2.64327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{875} (782, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 875,\ (\ :1/2),\ -0.297 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32747 + 1.80345i\)
\(L(\frac12)\) \(\approx\) \(1.32747 + 1.80345i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 2.08i)T \)
good2 \( 1 + (-0.00351 + 0.00434i)T + (-0.415 - 1.95i)T^{2} \)
3 \( 1 + (-1.74 - 1.13i)T + (1.22 + 2.74i)T^{2} \)
11 \( 1 + (3.82 + 1.70i)T + (7.36 + 8.17i)T^{2} \)
13 \( 1 + (-0.121 - 0.769i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.213 - 4.07i)T + (-16.9 + 1.77i)T^{2} \)
19 \( 1 + (-3.90 - 0.830i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (-1.73 - 1.40i)T + (4.78 + 22.4i)T^{2} \)
29 \( 1 + (-5.75 - 1.86i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.52 + 2.27i)T + (3.24 + 30.8i)T^{2} \)
37 \( 1 + (9.24 + 3.54i)T + (27.4 + 24.7i)T^{2} \)
41 \( 1 + (-6.32 + 8.70i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (3.91 - 3.91i)T - 43iT^{2} \)
47 \( 1 + (-9.02 - 0.472i)T + (46.7 + 4.91i)T^{2} \)
53 \( 1 + (0.266 - 0.411i)T + (-21.5 - 48.4i)T^{2} \)
59 \( 1 + (0.130 - 1.24i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (11.8 - 1.24i)T + (59.6 - 12.6i)T^{2} \)
67 \( 1 + (-9.52 + 0.499i)T + (66.6 - 7.00i)T^{2} \)
71 \( 1 + (-3.35 + 10.3i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.52 - 9.18i)T + (-54.2 + 48.8i)T^{2} \)
79 \( 1 + (-4.84 + 4.36i)T + (8.25 - 78.5i)T^{2} \)
83 \( 1 + (-0.767 + 1.50i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (0.115 + 1.09i)T + (-87.0 + 18.5i)T^{2} \)
97 \( 1 + (-1.14 - 2.24i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.41973864521117054194848898191, −9.227068365474877648421162913542, −8.686096253608831544863473451061, −8.069692440910596037108295143582, −7.37687162788141779793063029747, −5.95954647372269723822384001864, −4.94763622078310457465814414689, −3.79976303707142306442985784096, −3.04636876534087219454034703181, −2.16261503177845025957073125607, 0.985723792103344057516779435041, 2.17038401963248018597484591954, 3.08257791381294243652265509621, 4.75914177343307706144007888105, 5.29961149302656417540249251212, 6.77279087400716213837043916784, 7.39353724737120522773948119537, 8.047776068180542100650746335058, 9.024170578796002195437677987740, 9.963287921729922407012137305636

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