Properties

Label 2-475-19.9-c1-0-5
Degree $2$
Conductor $475$
Sign $0.853 - 0.520i$
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  + (−0.212 − 1.20i)2-s + (−0.616 + 0.517i)3-s + (0.477 − 0.173i)4-s + (0.752 + 0.631i)6-s + (−1.89 + 3.28i)7-s + (−1.53 − 2.65i)8-s + (−0.408 + 2.31i)9-s + (0.618 + 1.07i)11-s + (−0.204 + 0.353i)12-s + (2.64 + 2.22i)13-s + (4.35 + 1.58i)14-s + (−2.08 + 1.75i)16-s + (0.522 + 2.96i)17-s + 2.87·18-s + (4.28 − 0.793i)19-s + ⋯
L(s)  = 1  + (−0.149 − 0.850i)2-s + (−0.355 + 0.298i)3-s + (0.238 − 0.0868i)4-s + (0.307 + 0.257i)6-s + (−0.716 + 1.24i)7-s + (−0.541 − 0.937i)8-s + (−0.136 + 0.772i)9-s + (0.186 + 0.323i)11-s + (−0.0589 + 0.102i)12-s + (0.734 + 0.616i)13-s + (1.16 + 0.423i)14-s + (−0.522 + 0.438i)16-s + (0.126 + 0.718i)17-s + 0.677·18-s + (0.983 − 0.181i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.853 - 0.520i$
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: \((2,\ 475,\ (\ :1/2),\ 0.853 - 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04111 + 0.292553i\)
\(L(\frac12)\) \(\approx\) \(1.04111 + 0.292553i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.28 + 0.793i)T \)
good2 \( 1 + (0.212 + 1.20i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (0.616 - 0.517i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (1.89 - 3.28i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.618 - 1.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.64 - 2.22i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.522 - 2.96i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (5.77 - 2.10i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.744 - 4.22i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.55 - 4.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.13T + 37T^{2} \)
41 \( 1 + (-4.08 + 3.42i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-8.57 - 3.12i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.26 + 7.19i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.13 + 1.13i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.141 + 0.804i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (6.01 - 2.18i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.175 - 0.995i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (12.8 + 4.67i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (8.47 - 7.11i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (1.06 - 0.889i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.26 + 2.18i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.06 - 1.73i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.531 - 3.01i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.13597432343996989137592996752, −10.29931521164967200387548358703, −9.495545652458064427519271622006, −8.774882623979395957809663273963, −7.41207767202683635981565314476, −6.15969446494041285978369315484, −5.61060641108885200857129624287, −4.05356439089424308889300872895, −2.85664278660005360883438428203, −1.75330520148037283447066488004, 0.72980733325871370278437854785, 2.99100487930616743922388975096, 4.09026847780073217263639521223, 5.95568604858186337093431584259, 6.12178498755056635662539286705, 7.35472986487707922795359214613, 7.75407086661207274782857263475, 9.071942314296396672681531061402, 9.942431498913082774283893052915, 11.05348314716044448829130844713

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