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Properties

Label 2-475-95.94-c2-0-28
Degree $2$
Conductor $475$
Sign $-0.0605 + 0.998i$
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  − 2.36·2-s − 3.55·3-s + 1.59·4-s + 8.41·6-s − 11.0i·7-s + 5.68·8-s + 3.64·9-s + 16.7·11-s − 5.67·12-s − 14.3·13-s + 26.2i·14-s − 19.8·16-s + 20.2i·17-s − 8.61·18-s + (16.4 + 9.51i)19-s + ⋯
L(s)  = 1  − 1.18·2-s − 1.18·3-s + 0.399·4-s + 1.40·6-s − 1.58i·7-s + 0.710·8-s + 0.404·9-s + 1.52·11-s − 0.472·12-s − 1.10·13-s + 1.87i·14-s − 1.23·16-s + 1.18i·17-s − 0.478·18-s + (0.865 + 0.500i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0605 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0605 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.0605 + 0.998i$
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: \((2,\ 475,\ (\ :1),\ -0.0605 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4508197842\)
\(L(\frac12)\) \(\approx\) \(0.4508197842\)
\(L(2)\) 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 \( 1 + (-16.4 - 9.51i)T \)
good2 \( 1 + 2.36T + 4T^{2} \)
3 \( 1 + 3.55T + 9T^{2} \)
7 \( 1 + 11.0iT - 49T^{2} \)
11 \( 1 - 16.7T + 121T^{2} \)
13 \( 1 + 14.3T + 169T^{2} \)
17 \( 1 - 20.2iT - 289T^{2} \)
23 \( 1 + 13.5iT - 529T^{2} \)
29 \( 1 - 30.3iT - 841T^{2} \)
31 \( 1 + 16.1iT - 961T^{2} \)
37 \( 1 - 51.2T + 1.36e3T^{2} \)
41 \( 1 + 74.6iT - 1.68e3T^{2} \)
43 \( 1 - 6.23iT - 1.84e3T^{2} \)
47 \( 1 - 44.0iT - 2.20e3T^{2} \)
53 \( 1 - 30.8T + 2.80e3T^{2} \)
59 \( 1 + 73.0iT - 3.48e3T^{2} \)
61 \( 1 + 17.7T + 3.72e3T^{2} \)
67 \( 1 + 20.5T + 4.48e3T^{2} \)
71 \( 1 + 7.48iT - 5.04e3T^{2} \)
73 \( 1 + 29.2iT - 5.32e3T^{2} \)
79 \( 1 + 8.78iT - 6.24e3T^{2} \)
83 \( 1 + 63.7iT - 6.88e3T^{2} \)
89 \( 1 + 162. iT - 7.92e3T^{2} \)
97 \( 1 - 75.5T + 9.40e3T^{2} \)
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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

−10.48363001010870333173966984363, −9.862743162318042170546165423884, −8.930911162657661020342685219292, −7.71819009337559999532697327077, −7.04643830722312715722758010885, −6.17154099256719569813719452726, −4.75109659192554641847001833042, −3.89569503043748635059305971613, −1.42283654806903550381267656387, −0.46030256807071850279456719643, 0.985059870488110241304558019377, 2.58222988484769123349752316296, 4.63496653985967287473972515811, 5.46112188147644020834810357412, 6.48543128105231242749795870739, 7.39972153348537863270749150484, 8.581982232577797321821679644852, 9.495089151271398154451983060326, 9.699518852712096917046618236290, 11.19897467558423668299203799213

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