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Properties

Label 2-475-95.9-c1-0-19
Degree $2$
Conductor $475$
Sign $0.908 - 0.418i$
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  + (2.09 − 0.370i)2-s + (1.43 + 1.70i)3-s + (2.39 − 0.871i)4-s + (3.64 + 3.05i)6-s + (1.28 + 0.742i)7-s + (1.01 − 0.583i)8-s + (−0.342 + 1.94i)9-s + (−2.34 − 4.05i)11-s + (4.91 + 2.84i)12-s + (0.232 − 0.276i)13-s + (2.97 + 1.08i)14-s + (−1.99 + 1.67i)16-s + (−5.39 + 0.951i)17-s + 4.21i·18-s + (1.68 − 4.01i)19-s + ⋯
L(s)  = 1  + (1.48 − 0.261i)2-s + (0.827 + 0.986i)3-s + (1.19 − 0.435i)4-s + (1.48 + 1.24i)6-s + (0.486 + 0.280i)7-s + (0.357 − 0.206i)8-s + (−0.114 + 0.648i)9-s + (−0.705 − 1.22i)11-s + (1.42 + 0.819i)12-s + (0.0643 − 0.0767i)13-s + (0.795 + 0.289i)14-s + (−0.499 + 0.418i)16-s + (−1.30 + 0.230i)17-s + 0.992i·18-s + (0.386 − 0.922i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.908 - 0.418i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.908 - 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.65354 + 0.800742i\)
\(L(\frac12)\) \(\approx\) \(3.65354 + 0.800742i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 4.01i)T \)
good2 \( 1 + (-2.09 + 0.370i)T + (1.87 - 0.684i)T^{2} \)
3 \( 1 + (-1.43 - 1.70i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (-1.28 - 0.742i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.34 + 4.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.232 + 0.276i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (5.39 - 0.951i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-2.10 - 5.79i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.155 + 0.882i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.40 - 4.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.3iT - 37T^{2} \)
41 \( 1 + (4.01 - 3.36i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-2.46 + 6.78i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-10.7 - 1.88i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-2.23 - 6.12i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.70 - 9.65i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.20 - 0.803i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (8.71 + 1.53i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (6.02 + 2.19i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-1.83 - 2.19i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (1.58 - 1.32i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (5.33 + 3.08i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.54 - 2.13i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-4.64 + 0.819i)T + (91.1 - 33.1i)T^{2} \)
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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

−11.08031097172671553496277633274, −10.58072344932625601146823818601, −8.977166388245733523112110636372, −8.798556665417347860447193440098, −7.31226787821157421002778335324, −5.88411171383339618864430657684, −5.12261436088410499622085928889, −4.17853791548410267620319650173, −3.29709555839379287180654758313, −2.43770685442395939237668758451, 1.95687923481329987959609869755, 2.87975938184845569220030307829, 4.30677364175799079632261049119, 5.01460663347781071480885088671, 6.38376753425725065272358611340, 7.14607899205650509973119825341, 7.85483526484099295629067074722, 8.884717103669695229145029870707, 10.17063216376304815905080336927, 11.32772938359826946288478959668

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