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Properties

Label 2-475-95.49-c1-0-16
Degree $2$
Conductor $475$
Sign $0.645 - 0.764i$
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  + (1.03 + 0.595i)2-s + (2.63 + 1.52i)3-s + (−0.290 − 0.503i)4-s + (1.81 + 3.14i)6-s − 0.609i·7-s − 3.07i·8-s + (3.14 + 5.44i)9-s + 4.48·11-s − 1.77i·12-s + (−3.84 + 2.21i)13-s + (0.362 − 0.628i)14-s + (1.24 − 2.16i)16-s + (−2.51 − 1.45i)17-s + 7.48i·18-s + (−3.60 + 2.44i)19-s + ⋯
L(s)  = 1  + (0.729 + 0.421i)2-s + (1.52 + 0.879i)3-s + (−0.145 − 0.251i)4-s + (0.740 + 1.28i)6-s − 0.230i·7-s − 1.08i·8-s + (1.04 + 1.81i)9-s + 1.35·11-s − 0.511i·12-s + (−1.06 + 0.615i)13-s + (0.0969 − 0.167i)14-s + (0.312 − 0.540i)16-s + (−0.609 − 0.352i)17-s + 1.76i·18-s + (−0.827 + 0.562i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.645 - 0.764i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.645 - 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.75678 + 1.28044i\)
\(L(\frac12)\) \(\approx\) \(2.75678 + 1.28044i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 - 2.44i)T \)
good2 \( 1 + (-1.03 - 0.595i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-2.63 - 1.52i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + 0.609iT - 7T^{2} \)
11 \( 1 - 4.48T + 11T^{2} \)
13 \( 1 + (3.84 - 2.21i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.51 + 1.45i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.46 + 1.42i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.558 - 0.966i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.22T + 31T^{2} \)
37 \( 1 + 3.77iT - 37T^{2} \)
41 \( 1 + (-4.15 + 7.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.65 + 4.99i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.09 - 2.94i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.31 - 4.22i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.11 + 8.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.49 - 4.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.34 + 4.23i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.80 - 10.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.22 - 1.86i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.51 + 7.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.12iT - 83T^{2} \)
89 \( 1 + (-3.96 - 6.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.37 - 4.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.89781230892427785275894404195, −9.942288159756892135314820620994, −9.269431813719131737952677294863, −8.742757435057775289140238337761, −7.36875239281906525519477201221, −6.58612879393164128308352225790, −5.07314923875828238241322803368, −4.21280537474459352175075765443, −3.63187568442790082605954587462, −2.07521284210434386943903486288, 1.84672717579735951432355452135, 2.82352522189151544923548610558, 3.71294821077008778101358266536, 4.78091862313446491763995371100, 6.41681943835832160673014639543, 7.33727789645894239235975018799, 8.283720477149353972060622924276, 8.892210700928560199909110394985, 9.700832525172804084227345505472, 11.25644882287852348520778942625

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