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Properties

Label 2-475-95.49-c1-0-10
Degree $2$
Conductor $475$
Sign $0.846 + 0.532i$
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  + (−1.05 − 0.610i)2-s + (1.97 + 1.14i)3-s + (−0.253 − 0.439i)4-s + (−1.39 − 2.41i)6-s − 1.28i·7-s + 3.06i·8-s + (1.11 + 1.92i)9-s + 0.285·11-s − 1.15i·12-s + (4.33 − 2.5i)13-s + (−0.785 + 1.35i)14-s + (1.36 − 2.36i)16-s + (5.40 + 3.11i)17-s − 2.71i·18-s + (−2.92 − 3.22i)19-s + ⋯
L(s)  = 1  + (−0.748 − 0.431i)2-s + (1.14 + 0.659i)3-s + (−0.126 − 0.219i)4-s + (−0.569 − 0.987i)6-s − 0.485i·7-s + 1.08i·8-s + (0.370 + 0.641i)9-s + 0.0859·11-s − 0.334i·12-s + (1.20 − 0.693i)13-s + (−0.209 + 0.363i)14-s + (0.341 − 0.590i)16-s + (1.30 + 0.756i)17-s − 0.639i·18-s + (−0.671 − 0.740i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.846 + 0.532i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.846 + 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32236 - 0.381355i\)
\(L(\frac12)\) \(\approx\) \(1.32236 - 0.381355i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.92 + 3.22i)T \)
good2 \( 1 + (1.05 + 0.610i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.97 - 1.14i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + 1.28iT - 7T^{2} \)
11 \( 1 - 0.285T + 11T^{2} \)
13 \( 1 + (-4.33 + 2.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.40 - 3.11i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.53 + 2.61i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.642 - 1.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.22T + 31T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 + (-0.420 + 0.728i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.28 + 2.47i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.96 + 2.86i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.7 - 6.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.86 + 4.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.22 + 3.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.853 - 0.492i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.46 + 2.53i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.661 + 0.382i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.72 - 13.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.66iT - 83T^{2} \)
89 \( 1 + (8.01 + 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.1 + 5.87i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.48261206172751923644813095200, −10.13480595821379948806505530818, −9.061296218770222602790075641625, −8.537722271054482994534671067193, −7.83675431604603771166153610304, −6.30676042431863374878835548773, −5.02462883313954886470832796372, −3.79639384295229104006268568924, −2.82617042960674773856305883654, −1.20313498701264638439896798879, 1.45259554222653824464322057216, 2.99644297350544636708875777220, 3.96158581462814696809061145997, 5.73586482580326957970393264147, 6.92161490308140270906688225581, 7.65482960035276791627206931287, 8.417068975898890075396514785995, 9.017864923387313263007061588690, 9.678988065313151131798904385688, 10.99414959220513975966886082391

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