↑

Properties

Label 2-936-24.11-c1-0-1
Degree $2$
Conductor $936$
Sign $-0.478 - 0.878i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
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  + (−0.513 − 1.31i)2-s + (−1.47 + 1.35i)4-s − 0.398·5-s + 2.67i·7-s + (2.53 + 1.24i)8-s + (0.204 + 0.524i)10-s − 0.326i·11-s + i·13-s + (3.52 − 1.37i)14-s + (0.340 − 3.98i)16-s − 4.09i·17-s − 6.71·19-s + (0.586 − 0.538i)20-s + (−0.430 + 0.167i)22-s − 2.62·23-s + ⋯
L(s)  = 1  + (−0.362 − 0.931i)2-s + (−0.736 + 0.676i)4-s − 0.178·5-s + 1.01i·7-s + (0.897 + 0.440i)8-s + (0.0646 + 0.165i)10-s − 0.0984i·11-s + 0.277i·13-s + (0.941 − 0.366i)14-s + (0.0852 − 0.996i)16-s − 0.992i·17-s − 1.53·19-s + (0.131 − 0.120i)20-s + (−0.0917 + 0.0357i)22-s − 0.547·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.478 - 0.878i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.478 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113755 + 0.191467i\)
\(L(\frac12)\) \(\approx\) \(0.113755 + 0.191467i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.513 + 1.31i)T \)
3 \( 1 \)
13 \( 1 - iT \)
good5 \( 1 + 0.398T + 5T^{2} \)
7 \( 1 - 2.67iT - 7T^{2} \)
11 \( 1 + 0.326iT - 11T^{2} \)
17 \( 1 + 4.09iT - 17T^{2} \)
19 \( 1 + 6.71T + 19T^{2} \)
23 \( 1 + 2.62T + 23T^{2} \)
29 \( 1 + 3.53T + 29T^{2} \)
31 \( 1 - 3.68iT - 31T^{2} \)
37 \( 1 - 2.51iT - 37T^{2} \)
41 \( 1 - 0.433iT - 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 - 9.50T + 47T^{2} \)
53 \( 1 + 5.56T + 53T^{2} \)
59 \( 1 - 3.04iT - 59T^{2} \)
61 \( 1 + 3.06iT - 61T^{2} \)
67 \( 1 - 2.27T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 + 6.43T + 73T^{2} \)
79 \( 1 - 10.8iT - 79T^{2} \)
83 \( 1 - 0.495iT - 83T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 - 1.27T + 97T^{2} \)
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

−10.32062098244519622899858183851, −9.516939181678230202470085486880, −8.777708144902403953695788728127, −8.176139742266190361798401471828, −7.11160060868396509105065131948, −5.92316895402562532551944954600, −4.87646309162216061437880623854, −3.90568529282729356866715375250, −2.74739651674845167190285635675, −1.83209081235770141678628594971, 0.11460046222739264161967588441, 1.78834666918766485150392521474, 3.84306305380577911146055219876, 4.37506642574452304989835277358, 5.68729679558310355472666212184, 6.41696848441334678419582843859, 7.33723107702259844886562087817, 8.012860543173938349595856867733, 8.725292417450756265681286474652, 9.797834803904234539103342013570

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