↑

Properties

Label 2-837-93.92-c1-0-7
Degree $2$
Conductor $837$
Sign $-0.933 + 0.359i$
Analytic cond. $6.68347$
Root an. cond. $2.58524$
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  + 1.73i·2-s − 0.999·4-s + 3.46i·5-s + 2·7-s + 1.73i·8-s − 5.99·10-s − 6·11-s + 3.46i·14-s − 5·16-s + 3·17-s + 4·19-s − 3.46i·20-s − 10.3i·22-s − 6·23-s − 6.99·25-s + ⋯
L(s)  = 1  + 1.22i·2-s − 0.499·4-s + 1.54i·5-s + 0.755·7-s + 0.612i·8-s − 1.89·10-s − 1.80·11-s + 0.925i·14-s − 1.25·16-s + 0.727·17-s + 0.917·19-s − 0.774i·20-s − 2.21i·22-s − 1.25·23-s − 1.39·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.933 + 0.359i$
Analytic conductor: \(6.68347\)
Root analytic conductor: \(2.58524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (836, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 837,\ (\ :1/2),\ -0.933 + 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.258622 - 1.39189i\)
\(L(\frac12)\) \(\approx\) \(0.258622 - 1.39189i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 + (-2 - 5.19i)T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 5.19iT - 43T^{2} \)
47 \( 1 + 5.19iT - 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + 1.73iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 14T + 67T^{2} \)
71 \( 1 - 1.73iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 - 15.5iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 - 11T + 97T^{2} \)
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

−10.47184164577259266548818709096, −10.10012789257600988694109235924, −8.520839810989434246157083558131, −7.74255942600407098987049408045, −7.42955838027674491043567141789, −6.43149963698056450655360523081, −5.60339733845433149654577089609, −4.83794736382517457983396084500, −3.18998912768154095196843225304, −2.28477903326300886773087759210, 0.67931400258620233856188265920, 1.81274990696982997293394415470, 2.93011503847858510737829660415, 4.28857205528107958071506411829, 5.00316458534270154566652206252, 5.84058952130104413387825988706, 7.61054370239988270680216461347, 8.106166437703895138642426506825, 9.052745581040799837705992198639, 10.04095761900829772504785074318

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