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Properties

Label 1-475-475.206-r0-0-0
Degree $1$
Conductor $475$
Sign $-0.748 + 0.663i$
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  + (−0.719 − 0.694i)2-s + (−0.374 − 0.927i)3-s + (0.0348 + 0.999i)4-s + (−0.374 + 0.927i)6-s + (−0.5 + 0.866i)7-s + (0.669 − 0.743i)8-s + (−0.719 + 0.694i)9-s + (0.913 + 0.406i)11-s + (0.913 − 0.406i)12-s + (−0.241 − 0.970i)13-s + (0.961 − 0.275i)14-s + (−0.997 + 0.0697i)16-s + (−0.882 + 0.469i)17-s + 18-s + (0.990 + 0.139i)21-s + (−0.374 − 0.927i)22-s + ⋯
L(s)  = 1  + (−0.719 − 0.694i)2-s + (−0.374 − 0.927i)3-s + (0.0348 + 0.999i)4-s + (−0.374 + 0.927i)6-s + (−0.5 + 0.866i)7-s + (0.669 − 0.743i)8-s + (−0.719 + 0.694i)9-s + (0.913 + 0.406i)11-s + (0.913 − 0.406i)12-s + (−0.241 − 0.970i)13-s + (0.961 − 0.275i)14-s + (−0.997 + 0.0697i)16-s + (−0.882 + 0.469i)17-s + 18-s + (0.990 + 0.139i)21-s + (−0.374 − 0.927i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.748 + 0.663i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ -0.748 + 0.663i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05253337749 - 0.1385434168i\)
\(L(\frac12)\) \(\approx\) \(-0.05253337749 - 0.1385434168i\)
\(L(1)\) \(\approx\) \(0.4313615448 - 0.2404601602i\)
\(L(1)\) \(\approx\) \(0.4313615448 - 0.2404601602i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.719 - 0.694i)T \)
3 \( 1 + (-0.374 - 0.927i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.241 - 0.970i)T \)
17 \( 1 + (-0.882 + 0.469i)T \)
23 \( 1 + (0.559 - 0.829i)T \)
29 \( 1 + (-0.882 - 0.469i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.997 + 0.0697i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.882 - 0.469i)T \)
53 \( 1 + (0.0348 + 0.999i)T \)
59 \( 1 + (0.438 - 0.898i)T \)
61 \( 1 + (0.559 - 0.829i)T \)
67 \( 1 + (0.990 - 0.139i)T \)
71 \( 1 + (-0.615 + 0.788i)T \)
73 \( 1 + (-0.241 + 0.970i)T \)
79 \( 1 + (-0.374 - 0.927i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (-0.997 - 0.0697i)T \)
97 \( 1 + (0.990 + 0.139i)T \)
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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

−24.241808646473000915070855389833, −23.63306317819484543009187866319, −22.641737988903709613137662229952, −22.0507016735309753778527589099, −20.82928895840679153030550775588, −19.89927858260795205501044704615, −19.34168958916650879883929796640, −18.127247421048697956258432553285, −17.13951956776382978534038267398, −16.6651370657137804092603848963, −16.041333305891176722872326492, −15.02198139737859871986087153008, −14.249066108110095739646738823989, −13.36161806082102307096601790501, −11.60490200286928457590399062003, −11.078856589456583191938854494460, −10.003672040414086624391314226531, −9.32928575609214847452241507314, −8.64550329849904020680834625437, −7.0592214572885186096590449523, −6.61953294644362502085627579857, −5.40476148479766796023590292347, −4.424099110634771590235667010418, −3.42006723370896710948496074773, −1.50174552266659996073792961980, 0.1093681537676066674508724041, 1.66371633453568931943032447992, 2.46904165148702125198357899699, 3.63439367932832187606482661607, 5.18372097952543026436181452236, 6.42129417504332205416141301925, 7.151687376860975905943411935497, 8.32464034518867977667764239458, 8.99172035752726310820232725992, 10.08275227742798321887978756880, 11.14072508542218159627440048526, 11.89160465161031885732242272297, 12.75901863495594029312712255725, 13.11909842961613971592312353481, 14.627929756454048465485894803305, 15.70510375332290552026568538256, 16.91028249187050615497791339901, 17.40002407287870745384281345058, 18.36299185584991089844940321570, 18.9012970103946024446034442442, 19.82729224801404812703448520850, 20.325522209295528886645945736656, 21.81737197361860149161748653190, 22.31571189028457286039913334007, 23.02639780507621663590190077514

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