Properties

Label 2-475-95.94-c2-0-3
Degree $2$
Conductor $475$
Sign $-0.953 + 0.300i$
Analytic cond. $12.9428$
Root an. cond. $3.59761$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.151·2-s − 5.10·3-s − 3.97·4-s + 0.774·6-s + 6.35i·7-s + 1.20·8-s + 17.0·9-s + 10.0·11-s + 20.2·12-s − 3.79·13-s − 0.963i·14-s + 15.7·16-s + 13.4i·17-s − 2.58·18-s + (−2.99 + 18.7i)19-s + ⋯
L(s)  = 1  − 0.0758·2-s − 1.70·3-s − 0.994·4-s + 0.129·6-s + 0.907i·7-s + 0.151·8-s + 1.89·9-s + 0.913·11-s + 1.69·12-s − 0.291·13-s − 0.0688i·14-s + 0.982·16-s + 0.790i·17-s − 0.143·18-s + (−0.157 + 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.953 + 0.300i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (474, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ -0.953 + 0.300i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1275388791\)
\(L(\frac12)\) \(\approx\) \(0.1275388791\)
\(L(2)\) 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 \( 1 + (2.99 - 18.7i)T \)
good2 \( 1 + 0.151T + 4T^{2} \)
3 \( 1 + 5.10T + 9T^{2} \)
7 \( 1 - 6.35iT - 49T^{2} \)
11 \( 1 - 10.0T + 121T^{2} \)
13 \( 1 + 3.79T + 169T^{2} \)
17 \( 1 - 13.4iT - 289T^{2} \)
23 \( 1 - 16.7iT - 529T^{2} \)
29 \( 1 + 24.9iT - 841T^{2} \)
31 \( 1 + 46.3iT - 961T^{2} \)
37 \( 1 + 68.4T + 1.36e3T^{2} \)
41 \( 1 - 47.5iT - 1.68e3T^{2} \)
43 \( 1 - 43.4iT - 1.84e3T^{2} \)
47 \( 1 + 37.4iT - 2.20e3T^{2} \)
53 \( 1 + 9.48T + 2.80e3T^{2} \)
59 \( 1 - 87.7iT - 3.48e3T^{2} \)
61 \( 1 + 95.9T + 3.72e3T^{2} \)
67 \( 1 - 75.1T + 4.48e3T^{2} \)
71 \( 1 - 77.6iT - 5.04e3T^{2} \)
73 \( 1 + 120. iT - 5.32e3T^{2} \)
79 \( 1 + 12.8iT - 6.24e3T^{2} \)
83 \( 1 + 108. iT - 6.88e3T^{2} \)
89 \( 1 + 156. iT - 7.92e3T^{2} \)
97 \( 1 + 56.1T + 9.40e3T^{2} \)
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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

−11.49368870362971055367781313403, −10.34024380711915108394668921744, −9.690489816770062064791551166675, −8.733038059665671477521645127582, −7.61276640316944180631575206182, −6.19018644045519240898147616239, −5.78868318767380241595315792080, −4.77122015090623075296720961316, −3.83628618313700549613806075526, −1.49183147668948453301061458037, 0.088867154298231767800865522496, 1.11061678659313815261136263088, 3.74335568052747381930219640014, 4.77110200346724862419041587571, 5.28603237633863749862562849123, 6.69850881317140432972799985841, 7.13144854875599465201170964788, 8.681658117361564416736677423212, 9.575746544677976232054301830894, 10.55442340917689816271818153982

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