↑

Properties

Label 1-475-475.186-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.915 + 0.401i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.848 + 0.529i)2-s + (−0.961 − 0.275i)3-s + (0.438 − 0.898i)4-s + (0.961 − 0.275i)6-s + (−0.5 − 0.866i)7-s + (0.104 + 0.994i)8-s + (0.848 + 0.529i)9-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)12-s + (−0.0348 + 0.999i)13-s + (0.882 + 0.469i)14-s + (−0.615 − 0.788i)16-s + (−0.997 − 0.0697i)17-s − 18-s + (0.241 + 0.970i)21-s + (−0.961 − 0.275i)22-s + ⋯
L(s)  = 1  + (−0.848 + 0.529i)2-s + (−0.961 − 0.275i)3-s + (0.438 − 0.898i)4-s + (0.961 − 0.275i)6-s + (−0.5 − 0.866i)7-s + (0.104 + 0.994i)8-s + (0.848 + 0.529i)9-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)12-s + (−0.0348 + 0.999i)13-s + (0.882 + 0.469i)14-s + (−0.615 − 0.788i)16-s + (−0.997 − 0.0697i)17-s − 18-s + (0.241 + 0.970i)21-s + (−0.961 − 0.275i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.915 + 0.401i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.915 + 0.401i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04852707953 + 0.2312895099i\)
\(L(\frac12)\) \(\approx\) \(0.04852707953 + 0.2312895099i\)
\(L(1)\) \(\approx\) \(0.4756149037 + 0.06872882146i\)
\(L(1)\) \(\approx\) \(0.4756149037 + 0.06872882146i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.848 + 0.529i)T \)
3 \( 1 + (-0.961 - 0.275i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (-0.0348 + 0.999i)T \)
17 \( 1 + (-0.997 - 0.0697i)T \)
23 \( 1 + (0.990 + 0.139i)T \)
29 \( 1 + (0.997 - 0.0697i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.615 + 0.788i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.997 + 0.0697i)T \)
53 \( 1 + (-0.438 + 0.898i)T \)
59 \( 1 + (0.374 - 0.927i)T \)
61 \( 1 + (0.990 + 0.139i)T \)
67 \( 1 + (0.241 - 0.970i)T \)
71 \( 1 + (0.719 - 0.694i)T \)
73 \( 1 + (0.0348 + 0.999i)T \)
79 \( 1 + (-0.961 - 0.275i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (0.615 - 0.788i)T \)
97 \( 1 + (0.241 + 0.970i)T \)
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

−22.8930645372268302138144129288, −22.18199058965783873527176115480, −21.66331433132297395655013873008, −20.73824405681901051193742786530, −19.62107066625261824092702451356, −18.96029446339051573330561679923, −17.96068341784398784330994200819, −17.478570160987299012929136319567, −16.41484333484892152849194802533, −15.85750827698887944641646823655, −14.94687607788221203817705609506, −13.15825051380283938591328802783, −12.54273842094276368402109201860, −11.60232370748136234053846420261, −10.966001187929565331497348773645, −10.05426985508410358529950038247, −9.13149009022797918716127783754, −8.39100305315701661562241572181, −6.92119165311152870617656890936, −6.22958031023124452300894868541, −5.081641754696081492184507664576, −3.70227736574347591064454571417, −2.72208872399994880352979871519, −1.214508828355687871527194191183, −0.119908759865442106178956296921, 1.02961424390044120790380862961, 2.08067890696285930888843428207, 4.12828117747292601803451768573, 5.04063331485000932063618903479, 6.44476059216661919414248357532, 6.80219796186442907215226615238, 7.59607936992362866381101734818, 9.069535634188114944366131825817, 9.771364254014904415217057561634, 10.8024746085044239840707559183, 11.407904745184184912279235378737, 12.54192853095667626539378329683, 13.6090187126135031057811483402, 14.59968556961583813723728892657, 15.76461985079993456553956826250, 16.43717381130913501249240879807, 17.23376458435408316993744544861, 17.678365759543438373768528179028, 18.76722162560875248104836748788, 19.502190896096494399357470395153, 20.25633455088500567820780630891, 21.517946788151660730233884575574, 22.66207661560436865579057641082, 23.22738128084136857945478341338, 23.99683995210709244246714278231

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