Properties

Label 2-912-57.50-c1-0-13
Degree $2$
Conductor $912$
Sign $0.941 - 0.338i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 + 1.11i)3-s + (−2.85 + 1.64i)5-s + 4.86·7-s + (0.494 − 2.95i)9-s − 4.65i·11-s + (0.131 + 0.0760i)13-s + (1.92 − 5.37i)15-s + (−2.37 + 1.36i)17-s + (1.11 − 4.21i)19-s + (−6.43 + 5.44i)21-s + (4.87 + 2.81i)23-s + (2.92 − 5.06i)25-s + (2.65 + 4.46i)27-s + (3.39 − 5.87i)29-s − 2.83i·31-s + ⋯
L(s)  = 1  + (−0.763 + 0.646i)3-s + (−1.27 + 0.736i)5-s + 1.84·7-s + (0.164 − 0.986i)9-s − 1.40i·11-s + (0.0365 + 0.0210i)13-s + (0.497 − 1.38i)15-s + (−0.575 + 0.332i)17-s + (0.256 − 0.966i)19-s + (−1.40 + 1.18i)21-s + (1.01 + 0.587i)23-s + (0.585 − 1.01i)25-s + (0.511 + 0.859i)27-s + (0.629 − 1.09i)29-s − 0.508i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.941 - 0.338i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.941 - 0.338i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13356 + 0.197630i\)
\(L(\frac12)\) \(\approx\) \(1.13356 + 0.197630i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 1.11i)T \)
19 \( 1 + (-1.11 + 4.21i)T \)
good5 \( 1 + (2.85 - 1.64i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 4.86T + 7T^{2} \)
11 \( 1 + 4.65iT - 11T^{2} \)
13 \( 1 + (-0.131 - 0.0760i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.37 - 1.36i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.87 - 2.81i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.39 + 5.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.83iT - 31T^{2} \)
37 \( 1 - 5.14iT - 37T^{2} \)
41 \( 1 + (-0.166 - 0.287i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.12 - 3.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.83 - 4.52i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.486 - 0.842i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.30 - 5.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.39 + 2.42i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.73 - 2.15i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.75 + 13.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.13 - 5.42i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.38 + 4.84i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.12iT - 83T^{2} \)
89 \( 1 + (1.77 - 3.07i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.51 - 5.49i)T + (48.5 - 84.0i)T^{2} \)
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

−10.72870103594359869650776901289, −9.217486246141640740875584797103, −8.361149497459389695767045515201, −7.70035902603331874729118590027, −6.73006297389911665801292826456, −5.66320786048281104144562843221, −4.69861761715383048727719627269, −4.04083451952809822784801171813, −2.90848807286318108681813169868, −0.849728418084021186941086668153, 1.01700052709300284122424354034, 2.07937752547944517766490985653, 4.14916084472193311912604482215, 4.81501206466804747534800412847, 5.33932146313766419495743322481, 6.99934768143931622485118317787, 7.44842285228021335588966247073, 8.235273273551588513013678017654, 8.859063182882393754116937203166, 10.39105506472245549216582382440

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