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Properties

Label 2-475-19.9-c1-0-20
Degree $2$
Conductor $475$
Sign $0.677 - 0.735i$
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  + (0.428 + 2.43i)2-s + (2.17 − 1.82i)3-s + (−3.84 + 1.39i)4-s + (5.35 + 4.49i)6-s + (2.33 − 4.04i)7-s + (−2.57 − 4.46i)8-s + (0.874 − 4.95i)9-s + (−0.597 − 1.03i)11-s + (−5.79 + 10.0i)12-s + (0.684 + 0.574i)13-s + (10.8 + 3.93i)14-s + (3.47 − 2.91i)16-s + (1.04 + 5.91i)17-s + 12.4·18-s + (3.90 − 1.94i)19-s + ⋯
L(s)  = 1  + (0.302 + 1.71i)2-s + (1.25 − 1.05i)3-s + (−1.92 + 0.699i)4-s + (2.18 + 1.83i)6-s + (0.882 − 1.52i)7-s + (−0.911 − 1.57i)8-s + (0.291 − 1.65i)9-s + (−0.180 − 0.312i)11-s + (−1.67 + 2.89i)12-s + (0.189 + 0.159i)13-s + (2.89 + 1.05i)14-s + (0.869 − 0.729i)16-s + (0.253 + 1.43i)17-s + 2.92·18-s + (0.895 − 0.445i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.677 - 0.735i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.677 - 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15149 + 0.943864i\)
\(L(\frac12)\) \(\approx\) \(2.15149 + 0.943864i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.90 + 1.94i)T \)
good2 \( 1 + (-0.428 - 2.43i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (-2.17 + 1.82i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-2.33 + 4.04i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.597 + 1.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.684 - 0.574i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.04 - 5.91i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (6.03 - 2.19i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.294 + 1.67i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (3.08 - 5.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.25T + 37T^{2} \)
41 \( 1 + (1.29 - 1.08i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.824 - 0.299i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.29 - 7.34i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (7.01 - 2.55i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.640 - 3.63i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-7.02 + 2.55i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.60 - 9.13i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (0.151 + 0.0550i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.530 - 0.445i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-5.97 + 5.01i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (3.38 - 5.87i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (11.7 + 9.81i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.34 + 7.63i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.13633706105305916543596651332, −9.849686190327618816864390927841, −8.627318758244299401474419572035, −8.045039253545251318061696402098, −7.53709103079724086882218973019, −6.86322826643541530377401643987, −5.81836919673374765711624708294, −4.41648605945151716470675437181, −3.52322238108814349402899324339, −1.45641864782367025394663122869, 2.01789029193894051904115087915, 2.71251736349152461860777562089, 3.68359972108339417196047971730, 4.77144474454250930673184278567, 5.44359044300200852528106189013, 7.86817441250829716581150072382, 8.640751528518800907716990235634, 9.504527479573247170422073616683, 9.814030881226946824156086518513, 10.92743673497845169068276051792

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