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Properties

Label 2-475-19.11-c1-0-4
Degree $2$
Conductor $475$
Sign $-0.332 - 0.943i$
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 + (0.706 − 1.22i)3-s + (−1.35 − 2.34i)4-s + (1.53 + 2.65i)6-s − 1.76·7-s + 1.53·8-s + (0.502 + 0.869i)9-s + 1.83·11-s − 3.82·12-s + (1.30 + 2.25i)13-s + (1.91 − 3.31i)14-s + (1.03 − 1.79i)16-s + (−2.11 + 3.66i)17-s − 2.17·18-s + (4.01 + 1.68i)19-s + ⋯
L(s)  = 1  + (−0.767 + 1.32i)2-s + (0.407 − 0.706i)3-s + (−0.677 − 1.17i)4-s + (0.625 + 1.08i)6-s − 0.665·7-s + 0.544·8-s + (0.167 + 0.289i)9-s + 0.554·11-s − 1.10·12-s + (0.361 + 0.625i)13-s + (0.510 − 0.884i)14-s + (0.259 − 0.449i)16-s + (−0.513 + 0.889i)17-s − 0.513·18-s + (0.922 + 0.386i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.332 - 0.943i$
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: \((2,\ 475,\ (\ :1/2),\ -0.332 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.559565 + 0.790758i\)
\(L(\frac12)\) \(\approx\) \(0.559565 + 0.790758i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.01 - 1.68i)T \)
good2 \( 1 + (1.08 - 1.87i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.706 + 1.22i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 1.76T + 7T^{2} \)
11 \( 1 - 1.83T + 11T^{2} \)
13 \( 1 + (-1.30 - 2.25i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.11 - 3.66i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.10 - 1.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.56 - 6.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.303T + 31T^{2} \)
37 \( 1 - 3.90T + 37T^{2} \)
41 \( 1 + (4.11 - 7.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.17 - 2.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.62 - 6.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.31 + 9.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.02 + 10.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.26 + 9.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.51 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.91 - 10.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.58 + 7.94i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.94 - 6.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.93T + 83T^{2} \)
89 \( 1 + (-6.23 - 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.87 - 6.71i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.15501499934377497966790826623, −9.923556750481681911669351934166, −9.220038675561615281082765188978, −8.349167593663260191410131097308, −7.67945198878103214124009147914, −6.66217860335758983467637074133, −6.34009167159638753747782188585, −4.92075629529027030712818506919, −3.29457179697885986331146498158, −1.46887737928621608504058306615, 0.815684780045378418563873250216, 2.66264012564034306757586437517, 3.42622953729876189332582200801, 4.44923843118397330158869530158, 6.06595057458097954489068544412, 7.29146624386686707568840220628, 8.717836786705881439698714303819, 9.122547612664276662174494735363, 9.969866841873824415782196061932, 10.42558285594561067077771097569

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