↑

Properties

Label 2-936-8.5-c1-0-44
Degree $2$
Conductor $936$
Sign $-0.891 + 0.453i$
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  + (−0.221 − 1.39i)2-s + (−1.90 + 0.618i)4-s − 2.52i·5-s + 2.79·7-s + (1.28 + 2.52i)8-s + (−3.52 + 0.557i)10-s − 5.31i·11-s − i·13-s + (−0.618 − 3.90i)14-s + (3.23 − 2.35i)16-s + 4.35·17-s + 6.02i·19-s + (1.55 + 4.79i)20-s + (−7.42 + 1.17i)22-s − 0.568·23-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)2-s + (−0.951 + 0.309i)4-s − 1.12i·5-s + 1.05·7-s + (0.453 + 0.891i)8-s + (−1.11 + 0.176i)10-s − 1.60i·11-s − 0.277i·13-s + (−0.165 − 1.04i)14-s + (0.809 − 0.587i)16-s + 1.05·17-s + 1.38i·19-s + (0.348 + 1.07i)20-s + (−1.58 + 0.250i)22-s − 0.118·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.891 + 0.453i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.891 + 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.328500 - 1.36830i\)
\(L(\frac12)\) \(\approx\) \(0.328500 - 1.36830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.221 + 1.39i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 + 2.52iT - 5T^{2} \)
7 \( 1 - 2.79T + 7T^{2} \)
11 \( 1 + 5.31iT - 11T^{2} \)
17 \( 1 - 4.35T + 17T^{2} \)
19 \( 1 - 6.02iT - 19T^{2} \)
23 \( 1 + 0.568T + 23T^{2} \)
29 \( 1 + 2.68iT - 29T^{2} \)
31 \( 1 - 1.90T + 31T^{2} \)
37 \( 1 + 9.60iT - 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 9.48iT - 43T^{2} \)
47 \( 1 + 3.95T + 47T^{2} \)
53 \( 1 - 8.05iT - 53T^{2} \)
59 \( 1 - 6.19iT - 59T^{2} \)
61 \( 1 + 7.23iT - 61T^{2} \)
67 \( 1 - 10.0iT - 67T^{2} \)
71 \( 1 - 1.63T + 71T^{2} \)
73 \( 1 + 5.17T + 73T^{2} \)
79 \( 1 - 9.17T + 79T^{2} \)
83 \( 1 - 4.61iT - 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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

−9.812558840653766556106499096117, −8.694199464523085551745973107830, −8.382199296492541997285444562624, −7.68652397920144438189681557512, −5.72429784404812563131549004863, −5.29365412245374545185412639742, −4.16971847396744260068869098352, −3.28643488840537543677468399571, −1.73884798378739091189352570121, −0.77695252867396833865789747612, 1.65744336606196738957732437215, 3.17951892189321357392739428171, 4.63956630123590814036421672536, 5.04002962889726301649267776232, 6.48540477895079732343657075812, 6.99614504082446424847762110160, 7.73207494243277605918962098777, 8.498613793238107883637758966393, 9.683331769369341709728190715290, 10.12650832606568603985346096667

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