↑

Properties

Label 1-475-475.196-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

Related objects

Downloads

Learn more

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: $\chi_{475} (196, \cdot )$
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 \( 1 + (0.990 - 0.139i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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