↑

Properties

Label 2-475-95.94-c2-0-31
Degree $2$
Conductor $475$
Sign $0.812 - 0.582i$
Analytic cond. $12.9428$
Root an. cond. $3.59761$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.151·2-s + 5.10·3-s − 3.97·4-s + 0.774·6-s + 6.35i·7-s − 1.20·8-s + 17.0·9-s + 10.0·11-s − 20.2·12-s + 3.79·13-s + 0.963i·14-s + 15.7·16-s + 13.4i·17-s + 2.58·18-s + (−2.99 − 18.7i)19-s + ⋯
L(s)  = 1  + 0.0758·2-s + 1.70·3-s − 0.994·4-s + 0.129·6-s + 0.907i·7-s − 0.151·8-s + 1.89·9-s + 0.913·11-s − 1.69·12-s + 0.291·13-s + 0.0688i·14-s + 0.982·16-s + 0.790i·17-s + 0.143·18-s + (−0.157 − 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.812 - 0.582i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (474, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ 0.812 - 0.582i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.753178683\)
\(L(\frac12)\) \(\approx\) \(2.753178683\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (2.99 + 18.7i)T \)
good2 \( 1 - 0.151T + 4T^{2} \)
3 \( 1 - 5.10T + 9T^{2} \)
7 \( 1 - 6.35iT - 49T^{2} \)
11 \( 1 - 10.0T + 121T^{2} \)
13 \( 1 - 3.79T + 169T^{2} \)
17 \( 1 - 13.4iT - 289T^{2} \)
23 \( 1 - 16.7iT - 529T^{2} \)
29 \( 1 - 24.9iT - 841T^{2} \)
31 \( 1 - 46.3iT - 961T^{2} \)
37 \( 1 - 68.4T + 1.36e3T^{2} \)
41 \( 1 + 47.5iT - 1.68e3T^{2} \)
43 \( 1 - 43.4iT - 1.84e3T^{2} \)
47 \( 1 + 37.4iT - 2.20e3T^{2} \)
53 \( 1 - 9.48T + 2.80e3T^{2} \)
59 \( 1 + 87.7iT - 3.48e3T^{2} \)
61 \( 1 + 95.9T + 3.72e3T^{2} \)
67 \( 1 + 75.1T + 4.48e3T^{2} \)
71 \( 1 + 77.6iT - 5.04e3T^{2} \)
73 \( 1 + 120. iT - 5.32e3T^{2} \)
79 \( 1 - 12.8iT - 6.24e3T^{2} \)
83 \( 1 + 108. iT - 6.88e3T^{2} \)
89 \( 1 - 156. iT - 7.92e3T^{2} \)
97 \( 1 - 56.1T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.65049476302587512617153338875, −9.431304457606283866909918781560, −9.057561869398875283991826221635, −8.511707724270683190391952670772, −7.58080358193461731773532313378, −6.26919804834887637298460624637, −4.89484838747757908712223228358, −3.83191410643692579453402620418, −3.00082133550287443991655816689, −1.57861699288083896864776506687, 1.09277483421046480310374192738, 2.75000531180989876738081775545, 4.06177246896761252296482748838, 4.23671412219333157372030065481, 6.10242820452560221843035820850, 7.47363888433913291639783987734, 8.059047033395139099922884845416, 8.969258384894349955252157770978, 9.597183735277962562481229755315, 10.26362430019123358966089753741

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