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Properties

Label 1-475-475.198-r0-0-0
Degree $1$
Conductor $475$
Sign $-0.194 - 0.980i$
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  + (−0.207 − 0.978i)2-s + (0.994 + 0.104i)3-s + (−0.913 + 0.406i)4-s + (−0.104 − 0.994i)6-s − i·7-s + (0.587 + 0.809i)8-s + (0.978 + 0.207i)9-s + (0.309 − 0.951i)11-s + (−0.951 + 0.309i)12-s + (−0.207 + 0.978i)13-s + (−0.978 + 0.207i)14-s + (0.669 − 0.743i)16-s + (0.406 − 0.913i)17-s − i·18-s + (0.104 − 0.994i)21-s + (−0.994 − 0.104i)22-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (0.994 + 0.104i)3-s + (−0.913 + 0.406i)4-s + (−0.104 − 0.994i)6-s − i·7-s + (0.587 + 0.809i)8-s + (0.978 + 0.207i)9-s + (0.309 − 0.951i)11-s + (−0.951 + 0.309i)12-s + (−0.207 + 0.978i)13-s + (−0.978 + 0.207i)14-s + (0.669 − 0.743i)16-s + (0.406 − 0.913i)17-s − i·18-s + (0.104 − 0.994i)21-s + (−0.994 − 0.104i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.194 - 0.980i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (198, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ -0.194 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.015735227 - 1.236382351i\)
\(L(\frac12)\) \(\approx\) \(1.015735227 - 1.236382351i\)
\(L(1)\) \(\approx\) \(1.067686662 - 0.6747748711i\)
\(L(1)\) \(\approx\) \(1.067686662 - 0.6747748711i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.207 - 0.978i)T \)
3 \( 1 + (0.994 + 0.104i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.207 + 0.978i)T \)
17 \( 1 + (0.406 - 0.913i)T \)
23 \( 1 + (-0.743 + 0.669i)T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.406 - 0.913i)T \)
53 \( 1 + (-0.406 - 0.913i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (0.994 - 0.104i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (-0.207 - 0.978i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (-0.587 - 0.809i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (-0.994 - 0.104i)T \)
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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

−24.47949717411885008700059532677, −23.39054743084126580162341781074, −22.42810209290938310046067904655, −21.65903104146648582329383749749, −20.558501908493962690787638303690, −19.56782721767736055980362408625, −18.93636269311982492893049372491, −17.95236080544354622538361201760, −17.40466379681749571726608814387, −15.92968257212085512527562681030, −15.44436513333095355385303310215, −14.6391126605408611466415033241, −14.06819571143730600919512949132, −12.67529582729322576746533589642, −12.41393530427259010130450635436, −10.347779146701530673032013265508, −9.73287130217138644662004365272, −8.640286337204306019784783597808, −8.18601663092744215354713648368, −7.149547182354920503533595912831, −6.186887474901863555978382739425, −5.07508282290140312507764240217, −4.02629862209201361530326073421, −2.749771026752173602957363014380, −1.482520139603422628422844637436, 0.96098872127688190589073946659, 2.11744906493823386195960991554, 3.282988481767882848269714670, 3.95699323186269313863137476776, 4.92700212196646619491079301756, 6.738795261202925570353034654309, 7.81407449601807702040909628906, 8.57876061056608560199655544001, 9.61383604213040402482395610433, 10.13247333452782966978163708985, 11.30358547552782403430151227313, 12.069519949006511442644135753335, 13.47941224268369185785351556058, 13.77486150783578442763423615947, 14.46811957381069053345167125328, 15.97805692120192873562987268721, 16.75845227184325379864655075492, 17.78837011614402909644176610902, 18.94338562990685649684214380719, 19.32379090728110154961781419160, 20.18500053557771430123297023556, 20.926770894144785764162213556418, 21.55310191668929432405139738168, 22.48526044455109889525092241437, 23.57999609948034722530375944021

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