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Properties

Label 2-475-19.11-c1-0-13
Degree $2$
Conductor $475$
Sign $0.989 - 0.146i$
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  + (0.155 − 0.269i)2-s + (0.514 − 0.891i)3-s + (0.951 + 1.64i)4-s + (−0.160 − 0.277i)6-s + 3.28·7-s + 1.21·8-s + (0.969 + 1.67i)9-s − 5.16·11-s + 1.95·12-s + (1.76 + 3.06i)13-s + (0.510 − 0.883i)14-s + (−1.71 + 2.96i)16-s + (−0.504 + 0.874i)17-s + 0.603·18-s + (2.42 − 3.62i)19-s + ⋯
L(s)  = 1  + (0.109 − 0.190i)2-s + (0.297 − 0.514i)3-s + (0.475 + 0.824i)4-s + (−0.0653 − 0.113i)6-s + 1.23·7-s + 0.429·8-s + (0.323 + 0.559i)9-s − 1.55·11-s + 0.565·12-s + (0.490 + 0.849i)13-s + (0.136 − 0.236i)14-s + (−0.428 + 0.742i)16-s + (−0.122 + 0.211i)17-s + 0.142·18-s + (0.555 − 0.831i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.989 - 0.146i$
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: \((2,\ 475,\ (\ :1/2),\ 0.989 - 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00058 + 0.147218i\)
\(L(\frac12)\) \(\approx\) \(2.00058 + 0.147218i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.42 + 3.62i)T \)
good2 \( 1 + (-0.155 + 0.269i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.514 + 0.891i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 3.28T + 7T^{2} \)
11 \( 1 + 5.16T + 11T^{2} \)
13 \( 1 + (-1.76 - 3.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.504 - 0.874i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.83 + 6.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.01 - 3.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.60T + 31T^{2} \)
37 \( 1 - 6.48T + 37T^{2} \)
41 \( 1 + (-3.40 + 5.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.15 + 5.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.92 + 3.32i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.55 + 6.16i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.73 - 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.06 + 5.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.59 + 9.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.227 + 0.394i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.06 + 3.57i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.44 - 2.50i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.50T + 83T^{2} \)
89 \( 1 + (-3.56 - 6.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.41 - 9.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.86823690174494398365189872506, −10.68045350644073893581626534364, −8.923387058130292895547825483928, −8.035168817810422737299512588345, −7.65348455102235334405070065600, −6.67623642801789357102862226930, −5.13301105467250053010815389269, −4.25384563231141059121725535095, −2.65104240519539155850406982814, −1.86639212299655770787017072433, 1.39551652306263623325141105058, 2.89784213070802412340675698576, 4.38383642057555566271134348057, 5.35395011925648556211977415357, 6.04945101508224644405423215787, 7.67742795396989195145752606347, 7.940209812471432824888955558583, 9.444965805474070574207145383346, 10.12314142471837672138831849052, 10.90468054882854126873713389160

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