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Properties

Label 2-475-19.11-c1-0-14
Degree $2$
Conductor $475$
Sign $0.981 - 0.189i$
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  + (−1.08 + 1.87i)2-s + (−1.47 + 2.55i)3-s + (−1.35 − 2.34i)4-s + (−3.20 − 5.54i)6-s + 0.591·7-s + 1.53·8-s + (−2.85 − 4.94i)9-s + 2.58·11-s + 7.99·12-s + (−3.43 − 5.94i)13-s + (−0.641 + 1.11i)14-s + (1.03 − 1.79i)16-s + (2.61 − 4.53i)17-s + 12.3·18-s + (−2.26 + 3.72i)19-s + ⋯
L(s)  = 1  + (−0.767 + 1.32i)2-s + (−0.852 + 1.47i)3-s + (−0.677 − 1.17i)4-s + (−1.30 − 2.26i)6-s + 0.223·7-s + 0.544·8-s + (−0.952 − 1.64i)9-s + 0.778·11-s + 2.30·12-s + (−0.952 − 1.64i)13-s + (−0.171 + 0.297i)14-s + (0.259 − 0.449i)16-s + (0.634 − 1.09i)17-s + 2.92·18-s + (−0.519 + 0.854i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.981 - 0.189i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.981 - 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233514 + 0.0222797i\)
\(L(\frac12)\) \(\approx\) \(0.233514 + 0.0222797i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.26 - 3.72i)T \)
good2 \( 1 + (1.08 - 1.87i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.47 - 2.55i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 0.591T + 7T^{2} \)
11 \( 1 - 2.58T + 11T^{2} \)
13 \( 1 + (3.43 + 5.94i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.61 + 4.53i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.45 + 2.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.52 + 6.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.81T + 31T^{2} \)
37 \( 1 + 4.82T + 37T^{2} \)
41 \( 1 + (3.11 - 5.39i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.18 - 3.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.27 + 2.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.79 - 8.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.46 + 2.53i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.16 + 2.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.15 + 3.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.74 + 11.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.21 + 7.29i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.93 - 5.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.02T + 83T^{2} \)
89 \( 1 + (1.85 + 3.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.26 - 2.18i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.63994572131221698983166694469, −9.851730236644133317268565437849, −9.447755446700107801281524442352, −8.292029889723161211591547786462, −7.45507788860271529393423490597, −6.22714176789876102204113146462, −5.49669475375315674111875247463, −4.77729724064049683418017150367, −3.39168940184628059638862076542, −0.20833379265042068267002087974, 1.51460073396632903357543777364, 2.05755558609525390689132929757, 3.82899261827583705117741769017, 5.39900321785720619239792652892, 6.64301845586023668846479348419, 7.25614137920916750281528064427, 8.490253122290367439964315343501, 9.245136516241329228820814902086, 10.33780934533785417938834378679, 11.35071469558657578430102098175

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