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Properties

Label 2-475-95.48-c1-0-13
Degree $2$
Conductor $475$
Sign $0.863 + 0.503i$
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  + (1.34 − 1.92i)2-s + (−0.165 + 1.89i)3-s + (−1.19 − 3.28i)4-s + (3.41 + 2.86i)6-s + (0.719 + 2.68i)7-s + (−3.38 − 0.906i)8-s + (−0.609 − 0.107i)9-s + (1.57 + 2.72i)11-s + (6.42 − 1.72i)12-s + (3.82 − 0.334i)13-s + (6.12 + 2.22i)14-s + (−0.935 + 0.784i)16-s + (−4.24 − 2.97i)17-s + (−1.02 + 1.02i)18-s + (1.72 − 4.00i)19-s + ⋯
L(s)  = 1  + (0.950 − 1.35i)2-s + (−0.0957 + 1.09i)3-s + (−0.597 − 1.64i)4-s + (1.39 + 1.17i)6-s + (0.271 + 1.01i)7-s + (−1.19 − 0.320i)8-s + (−0.203 − 0.0358i)9-s + (0.475 + 0.823i)11-s + (1.85 − 0.496i)12-s + (1.06 − 0.0927i)13-s + (1.63 + 0.595i)14-s + (−0.233 + 0.196i)16-s + (−1.02 − 0.720i)17-s + (−0.241 + 0.241i)18-s + (0.395 − 0.918i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.863 + 0.503i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.863 + 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27607 - 0.615236i\)
\(L(\frac12)\) \(\approx\) \(2.27607 - 0.615236i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 4.00i)T \)
good2 \( 1 + (-1.34 + 1.92i)T + (-0.684 - 1.87i)T^{2} \)
3 \( 1 + (0.165 - 1.89i)T + (-2.95 - 0.520i)T^{2} \)
7 \( 1 + (-0.719 - 2.68i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.57 - 2.72i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.82 + 0.334i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (4.24 + 2.97i)T + (5.81 + 15.9i)T^{2} \)
23 \( 1 + (-0.380 - 0.177i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (1.19 - 6.78i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.180 - 0.103i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.00 + 7.00i)T + 37iT^{2} \)
41 \( 1 + (-1.39 - 1.66i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.30 + 7.09i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (-4.91 - 7.02i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (-1.58 + 3.40i)T + (-34.0 - 40.6i)T^{2} \)
59 \( 1 + (0.351 + 1.99i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (9.55 - 3.47i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.34 - 0.938i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-1.12 + 3.09i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (12.4 + 1.08i)T + (71.8 + 12.6i)T^{2} \)
79 \( 1 + (-3.75 + 3.14i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (1.17 - 0.315i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (2.74 + 2.30i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (4.27 - 6.10i)T + (-33.1 - 91.1i)T^{2} \)
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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

−10.99582429769210975528373633366, −10.43321884409315438486405400961, −9.234094784467108364541788525067, −8.986754998742556729670178946356, −7.06190035735594986558165274843, −5.58847144675094911146594554463, −4.86638254278068684303149135330, −4.09152891564799032414534277200, −3.03850619475135796736763278207, −1.79450288429365448228111981262, 1.38207651260947193958198558171, 3.63946925708570428822745520844, 4.38559996676200515818593063806, 5.89426450600684017336244183939, 6.38778566083514419902191951880, 7.18521198891758699981612542832, 7.995731828328976931918997088757, 8.652372993366461217745889789625, 10.32176565019961809730215523408, 11.36365876577542226140858640522

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