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Properties

Label 2-475-19.7-c1-0-7
Degree $2$
Conductor $475$
Sign $-0.257 - 0.966i$
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  + (0.431 + 0.747i)2-s + (1.53 + 2.66i)3-s + (0.627 − 1.08i)4-s + (−1.32 + 2.30i)6-s − 0.566·7-s + 2.80·8-s + (−3.24 + 5.61i)9-s − 1.91·11-s + 3.86·12-s + (0.0972 − 0.168i)13-s + (−0.244 − 0.423i)14-s + (−0.0438 − 0.0760i)16-s + (2.64 + 4.58i)17-s − 5.59·18-s + (−2.36 + 3.65i)19-s + ⋯
L(s)  = 1  + (0.305 + 0.528i)2-s + (0.888 + 1.53i)3-s + (0.313 − 0.543i)4-s + (−0.542 + 0.939i)6-s − 0.214·7-s + 0.993·8-s + (−1.08 + 1.87i)9-s − 0.576·11-s + 1.11·12-s + (0.0269 − 0.0467i)13-s + (−0.0653 − 0.113i)14-s + (−0.0109 − 0.0190i)16-s + (0.642 + 1.11i)17-s − 1.31·18-s + (−0.543 + 0.839i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.257 - 0.966i$
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: \((2,\ 475,\ (\ :1/2),\ -0.257 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40835 + 1.83186i\)
\(L(\frac12)\) \(\approx\) \(1.40835 + 1.83186i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.36 - 3.65i)T \)
good2 \( 1 + (-0.431 - 0.747i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.53 - 2.66i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 0.566T + 7T^{2} \)
11 \( 1 + 1.91T + 11T^{2} \)
13 \( 1 + (-0.0972 + 0.168i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.64 - 4.58i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.68 + 2.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.36 + 7.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 0.955T + 37T^{2} \)
41 \( 1 + (5.02 + 8.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.46 + 4.27i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.41 + 7.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.10 - 7.10i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.85 + 3.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.75 + 3.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.02 + 3.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.59 + 4.49i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.30 - 7.45i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.31 + 5.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.51T + 83T^{2} \)
89 \( 1 + (1.68 - 2.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.59 - 13.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.65480735766685308064667140241, −10.39288113865843560408729062916, −9.700117514500822263709627148382, −8.483491013175672646245312676873, −7.905836640142327887627079744749, −6.43358482037000427754616725896, −5.46526190257482908782091616849, −4.55098067483899670421478898733, −3.61101857869628465214667276316, −2.27641312930943371682793972174, 1.35524937247830206127772435533, 2.73902262646111318230320050660, 3.13637581438554406600844331283, 4.85185989622309320707305252512, 6.51163171964537117209906037155, 7.16455025721099274216444148582, 7.937116863203389126630577483602, 8.659006373156005730385078587686, 9.810042619696132495963780606909, 11.12121180781366240667685374553

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