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Properties

Label 2-475-95.4-c1-0-19
Degree $2$
Conductor $475$
Sign $0.992 - 0.120i$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
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  + (0.105 − 0.289i)2-s + (1.61 − 0.285i)3-s + (1.45 + 1.22i)4-s + (0.0879 − 0.498i)6-s + (0.0772 + 0.0445i)7-s + (1.04 − 0.601i)8-s + (−0.279 + 0.101i)9-s + (1.68 + 2.91i)11-s + (2.71 + 1.56i)12-s + (0.209 + 0.0369i)13-s + (0.0210 − 0.0176i)14-s + (0.597 + 3.38i)16-s + (0.859 − 2.36i)17-s + 0.0916i·18-s + (−0.949 − 4.25i)19-s + ⋯
L(s)  = 1  + (0.0744 − 0.204i)2-s + (0.934 − 0.164i)3-s + (0.729 + 0.612i)4-s + (0.0358 − 0.203i)6-s + (0.0291 + 0.0168i)7-s + (0.368 − 0.212i)8-s + (−0.0931 + 0.0339i)9-s + (0.507 + 0.879i)11-s + (0.782 + 0.452i)12-s + (0.0581 + 0.0102i)13-s + (0.00562 − 0.00471i)14-s + (0.149 + 0.846i)16-s + (0.208 − 0.573i)17-s + 0.0215i·18-s + (−0.217 − 0.975i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.992 - 0.120i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.992 - 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29000 + 0.138366i\)
\(L(\frac12)\) \(\approx\) \(2.29000 + 0.138366i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (0.949 + 4.25i)T \)
good2 \( 1 + (-0.105 + 0.289i)T + (-1.53 - 1.28i)T^{2} \)
3 \( 1 + (-1.61 + 0.285i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (-0.0772 - 0.0445i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.68 - 2.91i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.209 - 0.0369i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.859 + 2.36i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-3.83 + 4.57i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (4.51 - 1.64i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (4.03 - 6.98i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.84iT - 37T^{2} \)
41 \( 1 + (0.523 + 2.96i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (1.57 + 1.87i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (2.60 + 7.15i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-5.39 + 6.43i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-9.80 - 3.56i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.757 + 0.635i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-3.41 - 9.37i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-4.73 + 3.97i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (15.5 - 2.73i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-0.178 - 1.01i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (15.5 + 8.96i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.113 - 0.646i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (5.75 - 15.7i)T + (-74.3 - 62.3i)T^{2} \)
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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

−11.16672612386781790221315967138, −10.15751095777921134304232648285, −8.981286533310173579960997453107, −8.457189259141209211715727946275, −7.16914390452650253367835471069, −6.93855182044058619425545694606, −5.22013731412556842192827160924, −3.87233971272990981388102756083, −2.86362237005142619888544769919, −1.92498570230433020668166374619, 1.57938373478043673648631201876, 2.94954606594553564655298731521, 3.94823997944979559770807233033, 5.60676963647857866782468204531, 6.19396771208629696620081235065, 7.47839195104085535338494193694, 8.223360879641328985275562115150, 9.235169014721184960944218842567, 9.951126132095398731190772699794, 11.12541709991361486740543551410

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