Properties

Label 2-475-95.9-c1-0-26
Degree $2$
Conductor $475$
Sign $-0.999 - 0.00688i$
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  + (1.44 − 0.253i)2-s + (−0.970 − 1.15i)3-s + (0.130 − 0.0473i)4-s + (−1.69 − 1.41i)6-s + (−3.52 − 2.03i)7-s + (−2.35 + 1.36i)8-s + (0.124 − 0.707i)9-s + (0.310 + 0.537i)11-s + (−0.180 − 0.104i)12-s + (−3.27 + 3.90i)13-s + (−5.59 − 2.03i)14-s + (−3.26 + 2.73i)16-s + (0.262 − 0.0462i)17-s − 1.05i·18-s + (−0.399 − 4.34i)19-s + ⋯
L(s)  = 1  + (1.01 − 0.179i)2-s + (−0.560 − 0.667i)3-s + (0.0650 − 0.0236i)4-s + (−0.690 − 0.579i)6-s + (−1.33 − 0.769i)7-s + (−0.833 + 0.481i)8-s + (0.0416 − 0.235i)9-s + (0.0936 + 0.162i)11-s + (−0.0522 − 0.0301i)12-s + (−0.908 + 1.08i)13-s + (−1.49 − 0.544i)14-s + (−0.815 + 0.684i)16-s + (0.0636 − 0.0112i)17-s − 0.247i·18-s + (−0.0917 − 0.995i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.999 - 0.00688i$
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: \((2,\ 475,\ (\ :1/2),\ -0.999 - 0.00688i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00204404 + 0.593349i\)
\(L(\frac12)\) \(\approx\) \(0.00204404 + 0.593349i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.399 + 4.34i)T \)
good2 \( 1 + (-1.44 + 0.253i)T + (1.87 - 0.684i)T^{2} \)
3 \( 1 + (0.970 + 1.15i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (3.52 + 2.03i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.310 - 0.537i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.27 - 3.90i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.262 + 0.0462i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (1.99 + 5.48i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.708 - 4.01i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.83iT - 37T^{2} \)
41 \( 1 + (3.43 - 2.88i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.615 + 1.69i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-11.3 - 2.00i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (0.862 + 2.37i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.154 + 0.876i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.03 - 0.742i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (13.8 + 2.44i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.54 - 0.563i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (1.15 + 1.37i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (3.94 - 3.30i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-11.9 - 6.89i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.000572 - 0.000480i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-10.2 + 1.80i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.81940307426460205774239797540, −9.633745740462703521650555581837, −8.994818001159293123429829522883, −7.31616147475638644220674549325, −6.67190171440743742930595051637, −5.94708555682220302306935916970, −4.61419975250112164417309952745, −3.80664067067790972099937748494, −2.51699871289661178111296078785, −0.26369108647101514758201053163, 2.83586228671926723115904975072, 3.76839378354001180188609026370, 4.96394112413613867825449175556, 5.68364208831406602874941436346, 6.28133090821737653320652541018, 7.66947829093748218359817661504, 9.000335083124350274634142876479, 9.947072808109142706254344567306, 10.30115680644584277285690053247, 11.88878892481190501928443765941

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