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Properties

Label 2-475-19.7-c1-0-3
Degree $2$
Conductor $475$
Sign $0.910 + 0.412i$
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  + (−1.25 − 2.17i)2-s + (0.610 + 1.05i)3-s + (−2.14 + 3.71i)4-s + (1.53 − 2.65i)6-s + 0.221·7-s + 5.72·8-s + (0.753 − 1.30i)9-s − 0.778·11-s − 5.23·12-s + (−2.5 + 4.33i)13-s + (−0.278 − 0.481i)14-s + (−2.89 − 5.01i)16-s + (3.53 + 6.12i)17-s − 3.77·18-s + (1.33 + 4.15i)19-s + ⋯
L(s)  = 1  + (−0.886 − 1.53i)2-s + (0.352 + 0.610i)3-s + (−1.07 + 1.85i)4-s + (0.625 − 1.08i)6-s + 0.0838·7-s + 2.02·8-s + (0.251 − 0.435i)9-s − 0.234·11-s − 1.51·12-s + (−0.693 + 1.20i)13-s + (−0.0743 − 0.128i)14-s + (−0.724 − 1.25i)16-s + (0.858 + 1.48i)17-s − 0.890·18-s + (0.305 + 0.952i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904484 - 0.195483i\)
\(L(\frac12)\) \(\approx\) \(0.904484 - 0.195483i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 - 4.15i)T \)
good2 \( 1 + (1.25 + 2.17i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.610 - 1.05i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 0.221T + 7T^{2} \)
11 \( 1 + 0.778T + 11T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.53 - 6.12i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.03 + 6.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.110 - 0.192i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 - 1.90T + 37T^{2} \)
41 \( 1 + (-3.61 - 6.26i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.64 - 6.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.39 - 2.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.19 + 3.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.39 - 2.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.29 + 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.28 - 9.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.92 - 8.52i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.03 + 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.792 + 1.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.52T + 83T^{2} \)
89 \( 1 + (1.57 - 2.71i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.18 + 5.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.72570370520519314144769957306, −10.00698133303412958156068315149, −9.506449086445786822578826774308, −8.606375649925596022542469409066, −7.86060123988430621937283087617, −6.41999319106522427040436988799, −4.59389301000982668979591950169, −3.79261435691656696779702123729, −2.72906864628016150800411087153, −1.37590370879508833567984685784, 0.851493773775849930987144135948, 2.80181047719129562755059651019, 5.02710603811206847707130621460, 5.49100037982058781935466097656, 6.98806444053600022878436202671, 7.45256236702996570759877284254, 7.984010942388622338837020732399, 9.085824456466484498327632318087, 9.778287908915256567996642701861, 10.69031879008068018147244449966

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