↑

Properties

Label 2-936-24.11-c1-0-36
Degree $2$
Conductor $936$
Sign $0.989 - 0.147i$
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  + (0.508 + 1.31i)2-s + (−1.48 + 1.34i)4-s + 2.44·5-s − 2.55i·7-s + (−2.52 − 1.27i)8-s + (1.24 + 3.22i)10-s − 5.76i·11-s − i·13-s + (3.36 − 1.29i)14-s + (0.397 − 3.98i)16-s − 1.56i·17-s + 5.82·19-s + (−3.62 + 3.28i)20-s + (7.61 − 2.93i)22-s + 2.69·23-s + ⋯
L(s)  = 1  + (0.359 + 0.933i)2-s + (−0.741 + 0.671i)4-s + 1.09·5-s − 0.964i·7-s + (−0.892 − 0.450i)8-s + (0.393 + 1.02i)10-s − 1.73i·11-s − 0.277i·13-s + (0.900 − 0.346i)14-s + (0.0993 − 0.995i)16-s − 0.379i·17-s + 1.33·19-s + (−0.810 + 0.733i)20-s + (1.62 − 0.625i)22-s + 0.561·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.989 - 0.147i$
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: \((2,\ 936,\ (\ :1/2),\ 0.989 - 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99932 + 0.148328i\)
\(L(\frac12)\) \(\approx\) \(1.99932 + 0.148328i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.508 - 1.31i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 - 2.44T + 5T^{2} \)
7 \( 1 + 2.55iT - 7T^{2} \)
11 \( 1 + 5.76iT - 11T^{2} \)
17 \( 1 + 1.56iT - 17T^{2} \)
19 \( 1 - 5.82T + 19T^{2} \)
23 \( 1 - 2.69T + 23T^{2} \)
29 \( 1 + 6.76T + 29T^{2} \)
31 \( 1 + 5.29iT - 31T^{2} \)
37 \( 1 - 1.74iT - 37T^{2} \)
41 \( 1 - 7.99iT - 41T^{2} \)
43 \( 1 - 2.92T + 43T^{2} \)
47 \( 1 + 3.40T + 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 - 9.66iT - 59T^{2} \)
61 \( 1 + 0.282iT - 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 - 1.05T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 - 7.55iT - 79T^{2} \)
83 \( 1 - 14.2iT - 83T^{2} \)
89 \( 1 + 7.27iT - 89T^{2} \)
97 \( 1 - 4.07T + 97T^{2} \)
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

−9.736694639884768541434927768361, −9.319120586299770608180515902985, −8.195605635850676970626377528204, −7.50864012744002254363139651358, −6.51840752698994616902397071815, −5.76677143553211112114101867239, −5.14010252240726624835677322697, −3.81621412935217306399524315756, −2.94587734654847326323669884278, −0.892765529944684921246317537706, 1.69923307585062416443393516666, 2.27168815766449404895889842885, 3.52968260281078921018977000924, 4.89519196958644049329796228062, 5.39649086677332087053402323021, 6.34916572177682974354358341458, 7.47187829353705723093557779059, 8.857951455089864333230876248954, 9.506231074202230684653577750266, 9.864982995247299246264351330500

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