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Properties

Label 1-475-475.91-r1-0-0
Degree $1$
Conductor $475$
Sign $0.343 + 0.939i$
Analytic cond. $51.0458$
Root an. cond. $51.0458$
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  + (0.374 − 0.927i)2-s + (−0.559 + 0.829i)3-s + (−0.719 − 0.694i)4-s + (0.559 + 0.829i)6-s + (−0.5 − 0.866i)7-s + (−0.913 + 0.406i)8-s + (−0.374 − 0.927i)9-s + (−0.978 − 0.207i)11-s + (0.978 − 0.207i)12-s + (0.615 − 0.788i)13-s + (−0.990 + 0.139i)14-s + (0.0348 + 0.999i)16-s + (−0.241 − 0.970i)17-s − 18-s + (0.997 + 0.0697i)21-s + (−0.559 + 0.829i)22-s + ⋯
L(s)  = 1  + (0.374 − 0.927i)2-s + (−0.559 + 0.829i)3-s + (−0.719 − 0.694i)4-s + (0.559 + 0.829i)6-s + (−0.5 − 0.866i)7-s + (−0.913 + 0.406i)8-s + (−0.374 − 0.927i)9-s + (−0.978 − 0.207i)11-s + (0.978 − 0.207i)12-s + (0.615 − 0.788i)13-s + (−0.990 + 0.139i)14-s + (0.0348 + 0.999i)16-s + (−0.241 − 0.970i)17-s − 18-s + (0.997 + 0.0697i)21-s + (−0.559 + 0.829i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.343 + 0.939i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.343 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01814212901 + 0.01267995544i\)
\(L(\frac12)\) \(\approx\) \(0.01814212901 + 0.01267995544i\)
\(L(1)\) \(\approx\) \(0.6207423989 - 0.3602413192i\)
\(L(1)\) \(\approx\) \(0.6207423989 - 0.3602413192i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.374 - 0.927i)T \)
3 \( 1 + (-0.559 + 0.829i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (0.615 - 0.788i)T \)
17 \( 1 + (-0.241 - 0.970i)T \)
23 \( 1 + (-0.882 + 0.469i)T \)
29 \( 1 + (0.241 - 0.970i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.0348 - 0.999i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.241 + 0.970i)T \)
53 \( 1 + (0.719 + 0.694i)T \)
59 \( 1 + (-0.848 + 0.529i)T \)
61 \( 1 + (-0.882 + 0.469i)T \)
67 \( 1 + (0.997 - 0.0697i)T \)
71 \( 1 + (-0.438 - 0.898i)T \)
73 \( 1 + (-0.615 - 0.788i)T \)
79 \( 1 + (-0.559 + 0.829i)T \)
83 \( 1 + (-0.104 + 0.994i)T \)
89 \( 1 + (-0.0348 + 0.999i)T \)
97 \( 1 + (0.997 + 0.0697i)T \)
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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

−23.37389517861847857423231152774, −23.02511578299332777831568952070, −21.78302203354982566247589200003, −21.47230398639499913686622957668, −19.85613715389593094935262881925, −18.74467334771761148780322322637, −18.24524140492499926507170484592, −17.510151010349304340037410829502, −16.27439345684482425150739579119, −15.99336244671539170426968042007, −14.78975060664873229343376389211, −13.85442742305288456650038897522, −12.86215016055640929417550150068, −12.50825805962176445356610285422, −11.437731401079580832932268584024, −10.18811536132808273762876126414, −8.783211019520561368542773400370, −8.18674051872832456828413435930, −7.03150619688552922128106096215, −6.27229318115287951916150647129, −5.5834238072861829318693616820, −4.56654222228423138394824542752, −3.12616341792656030255784457945, −1.8666859342484551976818145950, −0.007275262509087898840409778771, 0.82118536579779465926376423127, 2.663811914536806587246076499687, 3.58654728495829934097395445393, 4.42865395903240294767024723660, 5.44415112484897424169274815633, 6.234486378059743641625082204465, 7.79335820058693194619431028281, 9.10531766687334354463251269900, 10.05519142356134264402558059276, 10.52460202140469172316228178346, 11.363233367024713970841756855268, 12.28230330566281795501039961314, 13.40649168659846101896351680934, 13.87458443164249250898895028460, 15.35102501618186276696065764834, 15.79683199068351461725974588818, 16.98380704803099805175309117984, 17.87869022343686827311126606677, 18.669558919561990141797469190372, 19.866123455064062257909633531469, 20.61797746356115032035036342758, 21.01907965187759367090562227866, 22.23128396527066613928146388760, 22.7009139950914502757409788189, 23.421300149619097065845872914913

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