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Properties

Label 2-475-19.7-c1-0-0
Degree $2$
Conductor $475$
Sign $-0.813 + 0.582i$
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  + (1.14 + 1.97i)2-s + (−1.25 − 2.17i)3-s + (−1.61 + 2.79i)4-s + (2.86 − 4.96i)6-s − 3.50·7-s − 2.79·8-s + (−1.64 + 2.84i)9-s − 4.50·11-s + 8.07·12-s + (−2.5 + 4.33i)13-s + (−4.00 − 6.94i)14-s + (0.0316 + 0.0547i)16-s + (0.0793 + 0.137i)17-s − 7.50·18-s + (−4.26 − 0.920i)19-s + ⋯
L(s)  = 1  + (0.807 + 1.39i)2-s + (−0.723 − 1.25i)3-s + (−0.805 + 1.39i)4-s + (1.16 − 2.02i)6-s − 1.32·7-s − 0.987·8-s + (−0.547 + 0.948i)9-s − 1.35·11-s + 2.33·12-s + (−0.693 + 1.20i)13-s + (−1.07 − 1.85i)14-s + (0.00790 + 0.0136i)16-s + (0.0192 + 0.0333i)17-s − 1.76·18-s + (−0.977 − 0.211i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.813 + 0.582i$
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: \((2,\ 475,\ (\ :1/2),\ -0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0661374 - 0.205939i\)
\(L(\frac12)\) \(\approx\) \(0.0661374 - 0.205939i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.26 + 0.920i)T \)
good2 \( 1 + (-1.14 - 1.97i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.25 + 2.17i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 3.50T + 7T^{2} \)
11 \( 1 + 4.50T + 11T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.0793 - 0.137i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.579 + 1.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.75 + 3.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.28T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + (3.03 + 5.26i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.67 + 2.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.53 + 2.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.87 - 4.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.53 + 2.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.436 + 0.756i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.22 - 7.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.11 - 14.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.57 + 6.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.06 - 8.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.85T + 83T^{2} \)
89 \( 1 + (-0.556 + 0.963i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.809 - 1.40i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.04268872892607586847846794086, −10.81105518472560256017036040510, −9.617788465893041322026044962170, −8.324767151051877294475340294184, −7.38026574699299279418345046448, −6.78734650159260270731538044008, −6.19030055253175724192564695084, −5.35146348392470295361451525187, −4.18516220977094387832365994037, −2.46877666432954634516220727658, 0.10201020570546821500503383326, 2.68650354244794469068216419111, 3.40688307374145805359966193421, 4.56386455995862912981249748125, 5.27241373098238115341083553711, 6.14872634281463052378911103775, 7.82077198152384746434390818703, 9.456658743270087684607599663355, 10.02042717909538157871654471960, 10.55998703799575554561156955048

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