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Properties

Label 2-475-95.49-c1-0-25
Degree $2$
Conductor $475$
Sign $-0.933 - 0.357i$
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  + (−1.73 − i)3-s + (−1 − 1.73i)4-s − 4i·7-s + (0.499 + 0.866i)9-s + 3·11-s + 3.99i·12-s + (−1.73 + i)13-s + (−1.99 + 3.46i)16-s + (−5.19 − 3i)17-s + (3.5 + 2.59i)19-s + (−4 + 6.92i)21-s + 4.00i·27-s + (−6.92 + 4i)28-s + (−1.5 − 2.59i)29-s − 7·31-s + ⋯
L(s)  = 1  + (−0.999 − 0.577i)3-s + (−0.5 − 0.866i)4-s − 1.51i·7-s + (0.166 + 0.288i)9-s + 0.904·11-s + 1.15i·12-s + (−0.480 + 0.277i)13-s + (−0.499 + 0.866i)16-s + (−1.26 − 0.727i)17-s + (0.802 + 0.596i)19-s + (−0.872 + 1.51i)21-s + 0.769i·27-s + (−1.30 + 0.755i)28-s + (−0.278 − 0.482i)29-s − 1.25·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.933 - 0.357i$
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: \((2,\ 475,\ (\ :1/2),\ -0.933 - 0.357i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0895911 + 0.484422i\)
\(L(\frac12)\) \(\approx\) \(0.0895911 + 0.484422i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 2.59i)T \)
good2 \( 1 + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.73 + i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (1.73 - i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.46 + 2i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.19 + 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.5 - 12.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 + i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.92 - 4i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.92 + 4i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.63163414635788753246573052026, −9.741225888898888338033886572149, −8.930378203587420553245210504139, −7.32686579733638003288005296332, −6.81146538933171466891201736027, −5.86974380884192101539714078010, −4.81245366832967215775226340318, −3.88766642861698776903591404347, −1.51400027269807577686427754073, −0.35725981704408066408398955194, 2.46650914425931066872841915969, 3.88816202389305248278849273958, 4.93651899916904404292786676175, 5.68195054417327968144985517669, 6.76772306089893675247472780653, 8.034547444365295615058664271505, 9.102242436495408917251378557091, 9.398117280555942099229369450246, 10.92241704573646771005278980669, 11.47374567986888504526435561718

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