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Properties

Label 1-475-475.213-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.909 - 0.415i$
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  + (0.927 + 0.374i)2-s + (0.829 + 0.559i)3-s + (0.719 + 0.694i)4-s + (0.559 + 0.829i)6-s + (−0.866 + 0.5i)7-s + (0.406 + 0.913i)8-s + (0.374 + 0.927i)9-s + (−0.978 − 0.207i)11-s + (0.207 + 0.978i)12-s + (−0.788 − 0.615i)13-s + (−0.990 + 0.139i)14-s + (0.0348 + 0.999i)16-s + (−0.970 + 0.241i)17-s + i·18-s + (−0.997 − 0.0697i)21-s + (−0.829 − 0.559i)22-s + ⋯
L(s)  = 1  + (0.927 + 0.374i)2-s + (0.829 + 0.559i)3-s + (0.719 + 0.694i)4-s + (0.559 + 0.829i)6-s + (−0.866 + 0.5i)7-s + (0.406 + 0.913i)8-s + (0.374 + 0.927i)9-s + (−0.978 − 0.207i)11-s + (0.207 + 0.978i)12-s + (−0.788 − 0.615i)13-s + (−0.990 + 0.139i)14-s + (0.0348 + 0.999i)16-s + (−0.970 + 0.241i)17-s + i·18-s + (−0.997 − 0.0697i)21-s + (−0.829 − 0.559i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.909 - 0.415i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.909 - 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4379937367 + 2.011602779i\)
\(L(\frac12)\) \(\approx\) \(-0.4379937367 + 2.011602779i\)
\(L(1)\) \(\approx\) \(1.349791226 + 1.025551167i\)
\(L(1)\) \(\approx\) \(1.349791226 + 1.025551167i\)

Euler product

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

−23.212165970763945459356489196165, −22.2681325144616746227163223698, −21.41452165567835446878342145135, −20.42859719384427211534392556475, −19.9016391019376863887658767588, −19.15331637035822254020937244809, −18.359286325110761238691881378586, −17.05540673907100269508360076922, −15.749744513031793878751598645314, −15.337396053284618752365251075665, −14.02259103183484094968141456985, −13.666458260010418785768467814927, −12.73383647806915520084675214868, −12.15532331776214333158429938105, −10.854414235754551823689062471531, −9.89760427778721588329606026893, −9.08104034065993075260646560421, −7.4851614758668204093173153531, −7.02324537884800224813003010574, −5.93100780948286751893987351163, −4.61982958725658396738242848039, −3.643860666022320512207578320687, −2.6803807776289908334383003880, −1.86458566518511620158813158396, −0.29034469761641043330397409194, 2.40098737623239828251676845056, 2.79280349941449747087971114523, 3.96081406554918616336113838489, 4.92817564275486048093009028772, 5.86449855756110961516210299627, 7.01992284946345228561072375477, 8.02519517550912206730419696847, 8.82752262485800781309327887187, 10.0690829296963939312282740808, 10.82120139170037463903766787601, 12.26150052402103780060405994261, 12.95315799435053783494421621585, 13.70407310540154672077288982361, 14.65707245827037752703602175180, 15.56697520387373667721743915485, 15.833391740777896426283315550273, 16.86508948104339080639886179526, 18.09056143529811225203332255178, 19.35377223416681791631454262105, 19.96424390464323123312074480204, 20.88960043336062586285160043998, 21.649744852433571604597537604811, 22.323362657169941925243476914743, 23.03641964361182425316706954661, 24.36081972244679343132928549434

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