↑

Properties

Label 2-912-228.155-c0-0-0
Degree $2$
Conductor $912$
Sign $-0.174 - 0.984i$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
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  + (−0.939 + 0.342i)3-s + (−0.592 + 0.342i)7-s + (0.766 − 0.642i)9-s + (−0.673 + 1.85i)13-s + (0.5 + 0.866i)19-s + (0.439 − 0.524i)21-s + (−0.939 − 0.342i)25-s + (−0.500 + 0.866i)27-s + (0.766 + 1.32i)31-s + 1.28i·37-s − 1.96i·39-s + (−1.26 − 0.223i)43-s + (−0.266 + 0.460i)49-s + (−0.766 − 0.642i)57-s + (−0.0603 − 0.342i)61-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (−0.592 + 0.342i)7-s + (0.766 − 0.642i)9-s + (−0.673 + 1.85i)13-s + (0.5 + 0.866i)19-s + (0.439 − 0.524i)21-s + (−0.939 − 0.342i)25-s + (−0.500 + 0.866i)27-s + (0.766 + 1.32i)31-s + 1.28i·37-s − 1.96i·39-s + (−1.26 − 0.223i)43-s + (−0.266 + 0.460i)49-s + (−0.766 − 0.642i)57-s + (−0.0603 − 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.174 - 0.984i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :0),\ -0.174 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5676519790\)
\(L(\frac12)\) \(\approx\) \(0.5676519790\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.592 - 0.342i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.673 - 1.85i)T + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (0.173 - 0.984i)T^{2} \)
31 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.28iT - T^{2} \)
41 \( 1 + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (1.26 + 0.223i)T + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \)
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

−10.40093460254596845951264435833, −9.744232539833920245483035816073, −9.178381200331383527829680073330, −7.970428786066182643394717430301, −6.73133533384777036717130615326, −6.41063832375135149734103706592, −5.24014512089965221020245744504, −4.44810605505007429666808240124, −3.38393479511351351385290943884, −1.76660595903075830785642840647, 0.62677652499218229339179798872, 2.47292719545086264288687281225, 3.74216452616591693323322566386, 5.02693358736580162719837644662, 5.66900981762752182794564855466, 6.61907529509317754752414536934, 7.47598194528789517845040458112, 8.093044006194893385664065829596, 9.583555573817103334692704629531, 10.06998999023388928029833429134

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