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Properties

Label 1-475-475.306-r1-0-0
Degree $1$
Conductor $475$
Sign $0.831 - 0.555i$
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  + (−0.961 + 0.275i)2-s + (−0.990 − 0.139i)3-s + (0.848 − 0.529i)4-s + (0.990 − 0.139i)6-s + (−0.5 + 0.866i)7-s + (−0.669 + 0.743i)8-s + (0.961 + 0.275i)9-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (0.719 − 0.694i)13-s + (0.241 − 0.970i)14-s + (0.438 − 0.898i)16-s + (0.0348 − 0.999i)17-s − 18-s + (0.615 − 0.788i)21-s + (−0.990 − 0.139i)22-s + ⋯
L(s)  = 1  + (−0.961 + 0.275i)2-s + (−0.990 − 0.139i)3-s + (0.848 − 0.529i)4-s + (0.990 − 0.139i)6-s + (−0.5 + 0.866i)7-s + (−0.669 + 0.743i)8-s + (0.961 + 0.275i)9-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (0.719 − 0.694i)13-s + (0.241 − 0.970i)14-s + (0.438 − 0.898i)16-s + (0.0348 − 0.999i)17-s − 18-s + (0.615 − 0.788i)21-s + (−0.990 − 0.139i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.831 - 0.555i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (306, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.831 - 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7077847571 - 0.2145516877i\)
\(L(\frac12)\) \(\approx\) \(0.7077847571 - 0.2145516877i\)
\(L(1)\) \(\approx\) \(0.5485820856 + 0.02926708834i\)
\(L(1)\) \(\approx\) \(0.5485820856 + 0.02926708834i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.961 + 0.275i)T \)
3 \( 1 + (-0.990 - 0.139i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (0.719 - 0.694i)T \)
17 \( 1 + (0.0348 - 0.999i)T \)
23 \( 1 + (-0.997 - 0.0697i)T \)
29 \( 1 + (-0.0348 - 0.999i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.438 + 0.898i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.0348 + 0.999i)T \)
53 \( 1 + (-0.848 + 0.529i)T \)
59 \( 1 + (-0.559 - 0.829i)T \)
61 \( 1 + (-0.997 - 0.0697i)T \)
67 \( 1 + (0.615 + 0.788i)T \)
71 \( 1 + (0.374 + 0.927i)T \)
73 \( 1 + (-0.719 - 0.694i)T \)
79 \( 1 + (-0.990 - 0.139i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (-0.438 - 0.898i)T \)
97 \( 1 + (0.615 - 0.788i)T \)
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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

−23.7984980238799302402653619467, −22.79108378604674466208419774239, −21.84343469475337896869928127286, −21.21017954965459905887149831590, −20.08754454295066320882665156167, −19.3575600048915005402973915511, −18.5113344908022311809581882674, −17.58990537273929723102199913365, −16.831729710836138897958760893985, −16.37706251565991794965266955282, −15.518206891105923913259145233834, −14.10780345949496553838546204479, −12.93113099457570248970365045203, −12.047279396289922279876455645809, −11.200245107891936203399675752968, −10.52426504228603083036437646199, −9.702739200985494952806143581457, −8.750953446970909430149586271987, −7.53296390195198118715482576596, −6.511262088889733001556717538957, −6.07910793178023342835578025158, −4.20781813901232771780926598375, −3.55212642360018439584311409647, −1.67656682711014262472642070890, −0.80417822480495688972888643952, 0.4453420825067169048629814082, 1.56983255251919345819986287938, 2.88274176597404472760298310188, 4.58168081536721805767244711094, 5.92746226362004465529087153873, 6.24153671548949480996875955479, 7.35290693686650654344935467804, 8.38849368278841851357440735544, 9.533954588617559350113097464110, 10.071390322211549140535227890356, 11.334153246222511131690841155, 11.84381184709012326353713836141, 12.7569319426172977436024031560, 14.11044613028174565232214015399, 15.508148917657518542955900433084, 15.7756124542624180901810170118, 16.807562061995637717362745948045, 17.57751527570227417646903400816, 18.32673537943068200001041316721, 18.920902082638755632812054999771, 19.93903466351443480130911172439, 20.86892575389617419178585714072, 22.02318960609441982401588397362, 22.73414392806922742935393144802, 23.54321607184447439128086345272

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