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Properties

Label 2-475-19.9-c1-0-2
Degree $2$
Conductor $475$
Sign $0.993 + 0.114i$
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  + (−0.455 − 2.58i)2-s + (0.711 − 0.597i)3-s + (−4.58 + 1.66i)4-s + (−1.86 − 1.56i)6-s + (−1.91 + 3.31i)7-s + (3.77 + 6.53i)8-s + (−0.371 + 2.10i)9-s + (1.63 + 2.82i)11-s + (−2.26 + 3.92i)12-s + (0.278 + 0.234i)13-s + (9.44 + 3.43i)14-s + (7.67 − 6.44i)16-s + (0.118 + 0.673i)17-s + 5.60·18-s + (−3.86 − 2.01i)19-s + ⋯
L(s)  = 1  + (−0.321 − 1.82i)2-s + (0.410 − 0.344i)3-s + (−2.29 + 0.833i)4-s + (−0.761 − 0.639i)6-s + (−0.724 + 1.25i)7-s + (1.33 + 2.30i)8-s + (−0.123 + 0.701i)9-s + (0.492 + 0.853i)11-s + (−0.653 + 1.13i)12-s + (0.0773 + 0.0649i)13-s + (2.52 + 0.918i)14-s + (1.91 − 1.61i)16-s + (0.0288 + 0.163i)17-s + 1.32·18-s + (−0.886 − 0.461i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.993 + 0.114i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.993 + 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.722230 - 0.0416559i\)
\(L(\frac12)\) \(\approx\) \(0.722230 - 0.0416559i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.86 + 2.01i)T \)
good2 \( 1 + (0.455 + 2.58i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (-0.711 + 0.597i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (1.91 - 3.31i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.63 - 2.82i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.278 - 0.234i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.118 - 0.673i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (8.80 - 3.20i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.267 + 1.51i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.22 + 2.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.163T + 37T^{2} \)
41 \( 1 + (5.64 - 4.73i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-2.43 - 0.885i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.52 - 8.66i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-10.4 + 3.80i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.35 + 7.67i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.27 - 0.827i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.65 + 9.41i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-4.81 - 1.75i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-6.03 + 5.06i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (7.14 - 5.99i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.364 + 0.631i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-9.68 - 8.12i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.31 - 7.43i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.07923001348331899764986821454, −10.02316632864910803067873521982, −9.459258276835241781808750082756, −8.633188519285944453139291658904, −7.86621935164309199641097669977, −6.25729414595082048041825411114, −4.86687544222606251388026548121, −3.71261331702215584847098774674, −2.48575924123072752552945702603, −1.90941704157255583324722874398, 0.46672067278371074403379820270, 3.66219738202981555897219606789, 4.25312071411303174169118421650, 5.82617614551678745681053586896, 6.48888491123600884980980858003, 7.22943908566585556411456735350, 8.383870012999400459615607008956, 8.827844934010855239938721312333, 9.957501511669217007163737472598, 10.37628456371706820469283966940

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