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Properties

Label 2-475-19.7-c1-0-20
Degree $2$
Conductor $475$
Sign $0.634 + 0.772i$
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  + (0.740 + 1.28i)2-s + (−1.42 − 2.47i)3-s + (−0.0969 + 0.167i)4-s + (2.11 − 3.66i)6-s + 3.78·7-s + 2.67·8-s + (−2.58 + 4.46i)9-s − 5.59·11-s + 0.554·12-s + (2.45 − 4.24i)13-s + (2.80 + 4.85i)14-s + (2.17 + 3.76i)16-s + (−0.875 − 1.51i)17-s − 7.64·18-s + (0.636 − 4.31i)19-s + ⋯
L(s)  = 1  + (0.523 + 0.907i)2-s + (−0.824 − 1.42i)3-s + (−0.0484 + 0.0839i)4-s + (0.863 − 1.49i)6-s + 1.43·7-s + 0.945·8-s + (−0.860 + 1.48i)9-s − 1.68·11-s + 0.159·12-s + (0.680 − 1.17i)13-s + (0.749 + 1.29i)14-s + (0.543 + 0.941i)16-s + (−0.212 − 0.367i)17-s − 1.80·18-s + (0.145 − 0.989i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.634 + 0.772i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.634 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47844 - 0.698643i\)
\(L(\frac12)\) \(\approx\) \(1.47844 - 0.698643i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.636 + 4.31i)T \)
good2 \( 1 + (-0.740 - 1.28i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.42 + 2.47i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 3.78T + 7T^{2} \)
11 \( 1 + 5.59T + 11T^{2} \)
13 \( 1 + (-2.45 + 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.875 + 1.51i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.290 + 0.503i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.832 - 1.44i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.01T + 31T^{2} \)
37 \( 1 + 2.36T + 37T^{2} \)
41 \( 1 + (0.417 + 0.723i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.535 + 0.927i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.93 + 3.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.39 - 5.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.204 - 0.353i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.98 - 12.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.390 + 0.676i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.18 + 5.52i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.44 - 2.50i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.25 - 10.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + (8.92 - 15.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.49 - 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.89445256125376499333271916110, −10.55484598956289665252142216375, −8.385647402969993108862767708530, −7.78844546675599673446407372401, −7.22735317535734425190228952210, −6.15557072617719858144884677366, −5.31005872396000869418519707579, −4.87768897859918202606440339258, −2.43143630532974815149996914050, −1.01951605240884858035925876842, 1.88909681648723903661872198931, 3.43387120647993613350302785857, 4.52830940267000153082417463473, 4.88549911270949531664431440430, 5.99432023460742536101881522211, 7.68387311860502731134543162910, 8.518898453109765466997359978838, 9.913926883229787070968979513265, 10.53934560532373322445120284800, 11.20325056307454626961687435994

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