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Properties

Label 1-475-475.166-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

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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: $\chi_{475} (166, \cdot )$
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 \( 1 + (0.241 - 0.970i)T \)
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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

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

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