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Properties

Label 2-475-475.111-c1-0-27
Degree $2$
Conductor $475$
Sign $-0.402 + 0.915i$
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  + (−0.934 − 0.0653i)2-s + (−0.0140 − 0.401i)3-s + (−1.11 − 0.156i)4-s + (−0.112 − 2.23i)5-s + (−0.0131 + 0.376i)6-s + (0.363 + 0.628i)7-s + (2.86 + 0.608i)8-s + (2.83 − 0.198i)9-s + (−0.0406 + 2.09i)10-s + (0.0645 + 0.614i)11-s + (−0.0471 + 0.448i)12-s + (−0.613 − 0.909i)13-s + (−0.298 − 0.611i)14-s + (−0.895 + 0.0765i)15-s + (−0.475 − 0.136i)16-s + (0.426 − 1.05i)17-s + ⋯
L(s)  = 1  + (−0.660 − 0.0462i)2-s + (−0.00809 − 0.231i)3-s + (−0.555 − 0.0781i)4-s + (−0.0503 − 0.998i)5-s + (−0.00536 + 0.153i)6-s + (0.137 + 0.237i)7-s + (1.01 + 0.215i)8-s + (0.943 − 0.0660i)9-s + (−0.0128 + 0.662i)10-s + (0.0194 + 0.185i)11-s + (−0.0136 + 0.129i)12-s + (−0.170 − 0.252i)13-s + (−0.0796 − 0.163i)14-s + (−0.231 + 0.0197i)15-s + (−0.118 − 0.0340i)16-s + (0.103 − 0.256i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.402 + 0.915i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.402 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.410119 - 0.628681i\)
\(L(\frac12)\) \(\approx\) \(0.410119 - 0.628681i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.112 + 2.23i)T \)
19 \( 1 + (1.93 + 3.90i)T \)
good2 \( 1 + (0.934 + 0.0653i)T + (1.98 + 0.278i)T^{2} \)
3 \( 1 + (0.0140 + 0.401i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (-0.363 - 0.628i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0645 - 0.614i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (0.613 + 0.909i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (-0.426 + 1.05i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (2.62 + 2.53i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (3.68 + 9.11i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (-0.460 + 0.511i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (3.97 + 2.88i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-10.3 - 2.95i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (1.07 - 6.11i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.17 - 2.91i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (-1.21 - 0.170i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (-0.177 - 0.710i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (4.60 + 4.44i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (13.0 + 8.13i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (-4.41 - 2.34i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (1.32 - 1.95i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (0.393 + 11.2i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (5.97 - 6.63i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (-4.76 + 1.36i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (-1.51 + 0.949i)T + (42.5 - 87.1i)T^{2} \)
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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

−10.47113927420270218467202782259, −9.605109394786860134253000448920, −9.070285609510476036084739440180, −8.066033540219101615485874030371, −7.45616272315541731673883207453, −6.01024064466558264330181632114, −4.78304020119009129922291502520, −4.18776551205429548282440600586, −2.01303514811207018487180528354, −0.61927213709865219844295607746, 1.67216650614349612597549887253, 3.55890736927701379976646707743, 4.34778153814165577252488239125, 5.71351612240552910748544748249, 7.08068716436280608609960282695, 7.57739418043624637210576902869, 8.680032224884306706582593449288, 9.608976940947510656691224155854, 10.41441126672210041791137025933, 10.77542146957852520832313071789

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