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Properties

Label 2-475-95.64-c1-0-9
Degree $2$
Conductor $475$
Sign $0.248 - 0.968i$
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  + (−1.28 + 0.740i)2-s + (0.157 − 0.0908i)3-s + (0.0969 − 0.167i)4-s + (−0.134 + 0.232i)6-s − 1.30i·7-s − 2.67i·8-s + (−1.48 + 2.56i)9-s + 4.98·11-s − 0.0352i·12-s + (−0.351 − 0.203i)13-s + (0.965 + 1.67i)14-s + (2.17 + 3.76i)16-s + (−2.38 + 1.37i)17-s − 4.39i·18-s + (4.35 − 0.0955i)19-s + ⋯
L(s)  = 1  + (−0.907 + 0.523i)2-s + (0.0908 − 0.0524i)3-s + (0.0484 − 0.0839i)4-s + (−0.0549 + 0.0951i)6-s − 0.492i·7-s − 0.945i·8-s + (−0.494 + 0.856i)9-s + 1.50·11-s − 0.0101i·12-s + (−0.0975 − 0.0563i)13-s + (0.258 + 0.447i)14-s + (0.543 + 0.941i)16-s + (−0.577 + 0.333i)17-s − 1.03i·18-s + (0.999 − 0.0219i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.248 - 0.968i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.248 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.679374 + 0.527213i\)
\(L(\frac12)\) \(\approx\) \(0.679374 + 0.527213i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.35 + 0.0955i)T \)
good2 \( 1 + (1.28 - 0.740i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.157 + 0.0908i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + 1.30iT - 7T^{2} \)
11 \( 1 - 4.98T + 11T^{2} \)
13 \( 1 + (0.351 + 0.203i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.38 - 1.37i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-6.02 - 3.47i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.00 - 3.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.57T + 31T^{2} \)
37 \( 1 - 3.71iT - 37T^{2} \)
41 \( 1 + (-0.607 - 1.05i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.70 - 1.56i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.63 - 3.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.47 - 3.16i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.61 - 9.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.467 - 0.808i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.58 + 2.64i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.817 - 1.41i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.65 - 3.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.27 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.2iT - 83T^{2} \)
89 \( 1 + (-7.10 + 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.8 + 9.14i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.06145633427498977328959848800, −10.09755002871540069387488257958, −9.097487669167826107938867446833, −8.680036868157212128521032308979, −7.43752057395838732697569810199, −7.05514618259196389622480677351, −5.79771037144846773348120591941, −4.41183137924044296836354152350, −3.27181880841286001415966610954, −1.29173614852179622037082151659, 0.853422278832169112607622453644, 2.37287478737508932707016123438, 3.70537502943446461264781542777, 5.15479178345879010400032186209, 6.22169015375313443369583992891, 7.26334961687536130927653540569, 8.699743735897775999885061041559, 9.052977450823591130254577747900, 9.663538445127767360367600274292, 10.80028930130594039665620980681

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