↑

Properties

Label 1-475-475.119-r0-0-0
Degree $1$
Conductor $475$
Sign $-0.984 + 0.176i$
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  + (0.241 − 0.970i)2-s + (0.615 − 0.788i)3-s + (−0.882 − 0.469i)4-s + (−0.615 − 0.788i)6-s + (0.5 − 0.866i)7-s + (−0.669 + 0.743i)8-s + (−0.241 − 0.970i)9-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (−0.961 − 0.275i)13-s + (−0.719 − 0.694i)14-s + (0.559 + 0.829i)16-s + (−0.848 − 0.529i)17-s − 18-s + (−0.374 − 0.927i)21-s + (0.615 − 0.788i)22-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)2-s + (0.615 − 0.788i)3-s + (−0.882 − 0.469i)4-s + (−0.615 − 0.788i)6-s + (0.5 − 0.866i)7-s + (−0.669 + 0.743i)8-s + (−0.241 − 0.970i)9-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (−0.961 − 0.275i)13-s + (−0.719 − 0.694i)14-s + (0.559 + 0.829i)16-s + (−0.848 − 0.529i)17-s − 18-s + (−0.374 − 0.927i)21-s + (0.615 − 0.788i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.984 + 0.176i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ -0.984 + 0.176i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1358852027 - 1.527530267i\)
\(L(\frac12)\) \(\approx\) \(-0.1358852027 - 1.527530267i\)
\(L(1)\) \(\approx\) \(0.7051808312 - 1.064618950i\)
\(L(1)\) \(\approx\) \(0.7051808312 - 1.064618950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.241 - 0.970i)T \)
3 \( 1 + (0.615 - 0.788i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.961 - 0.275i)T \)
17 \( 1 + (-0.848 - 0.529i)T \)
23 \( 1 + (-0.438 - 0.898i)T \)
29 \( 1 + (0.848 - 0.529i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.559 + 0.829i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (-0.848 + 0.529i)T \)
53 \( 1 + (0.882 + 0.469i)T \)
59 \( 1 + (-0.997 + 0.0697i)T \)
61 \( 1 + (0.438 + 0.898i)T \)
67 \( 1 + (0.374 - 0.927i)T \)
71 \( 1 + (0.990 + 0.139i)T \)
73 \( 1 + (-0.961 + 0.275i)T \)
79 \( 1 + (-0.615 + 0.788i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (0.559 - 0.829i)T \)
97 \( 1 + (0.374 + 0.927i)T \)
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

−24.63797482349607850799612434312, −23.583624947327065543219162188967, −22.30688154555504969881592421691, −21.77081702951800860947966776244, −21.38191699261795309550845196543, −19.90643614919686349773511715724, −19.26389948675318705924672064966, −18.047569482482935129691403920713, −17.246116341231978357647934132010, −16.31637503919429607133806155240, −15.59728552421147642217330988096, −14.6248633706052855240126902332, −14.42236954005324026710553060224, −13.26401579101180412198800008091, −12.15949099585206026648599741636, −11.146197059793732242185890650381, −9.72445147077844965585455500750, −9.03223321959367254151392501290, −8.37641570372041547772237821399, −7.36195192572245263797209802944, −6.14723107527486664689861379118, −5.16933386447500842160391379395, −4.34618888606580133053395701998, −3.36653264761582672994302017313, −2.059046358922788063313663275936, 0.73040187315689870865921468969, 1.869402872692308231476049096740, 2.747536525208915040368825581343, 4.00703692690572749153686381107, 4.74009499118799215070068277625, 6.30604538549423712760482403027, 7.30497908936914205015964362174, 8.301361295320484120213159929, 9.30046534116974736180257862296, 10.11385537725622215157559083538, 11.28768880214408228479563805390, 12.052710170332306356418225086961, 12.87967948292486122054260561445, 13.75728187121075910332359631889, 14.417825314779222168970636021502, 15.07044359774700508437976350479, 16.87217929585136483712171986251, 17.69597909488772292185100122919, 18.3184683484225465121833908168, 19.45848989767294873012502263529, 20.05896337477926860872920496307, 20.41405270897709618457905195858, 21.590335197005822285387796367407, 22.54101750310419020382877372368, 23.27048106944362561225588152411

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