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Properties

Label 1-475-475.238-r0-0-0
Degree $1$
Conductor $475$
Sign $-0.667 - 0.744i$
Analytic cond. $2.20589$
Root an. cond. $2.20589$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.788 − 0.615i)2-s + (0.898 − 0.438i)3-s + (0.241 − 0.970i)4-s + (0.438 − 0.898i)6-s + (−0.866 + 0.5i)7-s + (−0.406 − 0.913i)8-s + (0.615 − 0.788i)9-s + (−0.978 − 0.207i)11-s + (−0.207 − 0.978i)12-s + (0.139 − 0.990i)13-s + (−0.374 + 0.927i)14-s + (−0.882 − 0.469i)16-s + (0.275 − 0.961i)17-s − i·18-s + (−0.559 + 0.829i)21-s + (−0.898 + 0.438i)22-s + ⋯
L(s)  = 1  + (0.788 − 0.615i)2-s + (0.898 − 0.438i)3-s + (0.241 − 0.970i)4-s + (0.438 − 0.898i)6-s + (−0.866 + 0.5i)7-s + (−0.406 − 0.913i)8-s + (0.615 − 0.788i)9-s + (−0.978 − 0.207i)11-s + (−0.207 − 0.978i)12-s + (0.139 − 0.990i)13-s + (−0.374 + 0.927i)14-s + (−0.882 − 0.469i)16-s + (0.275 − 0.961i)17-s − i·18-s + (−0.559 + 0.829i)21-s + (−0.898 + 0.438i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.667 - 0.744i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ -0.667 - 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9552511354 - 2.139125898i\)
\(L(\frac12)\) \(\approx\) \(0.9552511354 - 2.139125898i\)
\(L(1)\) \(\approx\) \(1.408564187 - 1.153610276i\)
\(L(1)\) \(\approx\) \(1.408564187 - 1.153610276i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.788 - 0.615i)T \)
3 \( 1 + (0.898 - 0.438i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (0.139 - 0.990i)T \)
17 \( 1 + (0.275 - 0.961i)T \)
23 \( 1 + (-0.529 + 0.848i)T \)
29 \( 1 + (0.961 - 0.275i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.882 + 0.469i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (-0.275 - 0.961i)T \)
53 \( 1 + (0.970 + 0.241i)T \)
59 \( 1 + (0.0348 + 0.999i)T \)
61 \( 1 + (0.848 + 0.529i)T \)
67 \( 1 + (-0.829 + 0.559i)T \)
71 \( 1 + (0.997 + 0.0697i)T \)
73 \( 1 + (0.139 + 0.990i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (-0.994 - 0.104i)T \)
89 \( 1 + (-0.882 + 0.469i)T \)
97 \( 1 + (0.829 + 0.559i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.96994612093315929547746457408, −23.44346666211174374579348302332, −22.41783206196092408222440943492, −21.5409268372008627832460197086, −20.97694877798481524019459171653, −20.036588006148183073030440628576, −19.245192820060040022469566378366, −18.12435306325665958434399930213, −16.817554079770591079650342385592, −16.14063963548077360188227305651, −15.56795198378373184628698492551, −14.486149671329882812517069416709, −13.91694134207793062592745676541, −13.00465200809268482134232672652, −12.38217737439959796386638365121, −10.8346578869382315955570489139, −10.00236460206257164104743833164, −8.842022421668283445519456992626, −8.01408776581529370270411301339, −7.07178461427277806313862283359, −6.16674299621347577957026895890, −4.82437534827777270891878818135, −4.02302122924991095011328779437, −3.146829177021933454948988129401, −2.13704927791420243937756321020, 0.88188812455066764008865637988, 2.544063473394946269710332189569, 2.86458324508433351561236114818, 3.954156608329161142978925403309, 5.347705002017378062781632333008, 6.17799465384430132418152917464, 7.34393275513095164823919878082, 8.366622125976650549608459825814, 9.64812536132010904417359872530, 10.07827177345176928812059906503, 11.4876590308258395117443963935, 12.3845581902267780701210060474, 13.251119004138264027145673466860, 13.5549655703522959476387370496, 14.79627120658963239950746044311, 15.50256140116586051883792221678, 16.16295075809533143609451077727, 18.115179237155745182830024535801, 18.49999294231930086228030437363, 19.55164733550181180814262629112, 20.04169591172953421605328718534, 20.989563329355732963908989432519, 21.62946755360016921203743062414, 22.75483434126327033769916767808, 23.343835402229382265498075914346

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