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Properties

Label 2-475-95.8-c1-0-21
Degree $2$
Conductor $475$
Sign $-0.691 + 0.722i$
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  + (−0.965 + 0.258i)2-s + (−0.567 − 2.11i)3-s + (−0.866 + 0.5i)4-s + (1.09 + 1.89i)6-s + (1.22 − 1.22i)7-s + (2.12 − 2.12i)8-s + (−1.56 + 0.903i)9-s + 1.19·11-s + (1.55 + 1.55i)12-s + (1.15 + 0.308i)13-s + (−0.866 + 1.49i)14-s + (−0.500 + 0.866i)16-s + (−0.982 − 3.66i)17-s + (1.27 − 1.27i)18-s + (−4.33 + 0.5i)19-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.327 − 1.22i)3-s + (−0.433 + 0.250i)4-s + (0.447 + 0.775i)6-s + (0.462 − 0.462i)7-s + (0.749 − 0.749i)8-s + (−0.521 + 0.301i)9-s + 0.359·11-s + (0.447 + 0.447i)12-s + (0.319 + 0.0856i)13-s + (−0.231 + 0.400i)14-s + (−0.125 + 0.216i)16-s + (−0.238 − 0.889i)17-s + (0.301 − 0.301i)18-s + (−0.993 + 0.114i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.691 + 0.722i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.245728 - 0.575452i\)
\(L(\frac12)\) \(\approx\) \(0.245728 - 0.575452i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.33 - 0.5i)T \)
good2 \( 1 + (0.965 - 0.258i)T + (1.73 - i)T^{2} \)
3 \( 1 + (0.567 + 2.11i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.22 + 1.22i)T - 7iT^{2} \)
11 \( 1 - 1.19T + 11T^{2} \)
13 \( 1 + (-1.15 - 0.308i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.982 + 3.66i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (-0.448 + 1.67i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.89 - 3.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.26iT - 31T^{2} \)
37 \( 1 + (5.92 + 5.92i)T + 37iT^{2} \)
41 \( 1 + (4.78 + 2.76i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.68 - 2.32i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (9.00 + 2.41i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.76 - 1.81i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.66 + 8.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.78 - 3.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.78 + 10.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-14.0 - 8.12i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.02 - 0.810i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.29 + 14.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.46 + 1.46i)T + 83iT^{2} \)
89 \( 1 + (-2.59 - 4.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.74 - 0.467i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.67895089659091783610054760338, −9.646209446385302819984917446528, −8.643896956473388390422127441590, −7.948130175954891156599721115439, −7.07092394682263247802769670438, −6.48169244841287674391163377373, −4.97005203410111126687345311342, −3.81723655247120487194574350543, −1.85618032101176680683344032391, −0.53014061352184757421243655357, 1.72937141468179000331392506983, 3.69342300462807491073454156570, 4.71388349191007941955259547081, 5.38847125824229069019377233707, 6.65355553867817534317633282918, 8.416513280482517132803562011740, 8.584645005070059924068122679951, 9.771274069650718682253372918669, 10.28625044834686248421259488887, 11.02002069189088004943941957377

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