↑

Properties

Label 2-816-12.11-c1-0-23
Degree $2$
Conductor $816$
Sign $-0.595 + 0.803i$
Analytic cond. $6.51579$
Root an. cond. $2.55260$
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  + (−1.39 − 1.03i)3-s − 2i·5-s − 2.78i·7-s + (0.870 + 2.87i)9-s + 3.50·11-s + 5.74·13-s + (−2.06 + 2.78i)15-s − i·17-s − 5.56i·19-s + (−2.87 + 3.87i)21-s − 6.90·23-s + 25-s + (1.75 − 4.89i)27-s + 5.48i·29-s − 3.50i·31-s + ⋯
L(s)  = 1  + (−0.803 − 0.595i)3-s − 0.894i·5-s − 1.05i·7-s + (0.290 + 0.956i)9-s + 1.05·11-s + 1.59·13-s + (−0.532 + 0.718i)15-s − 0.242i·17-s − 1.27i·19-s + (−0.626 + 0.844i)21-s − 1.44·23-s + 0.200·25-s + (0.336 − 0.941i)27-s + 1.01i·29-s − 0.628i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $-0.595 + 0.803i$
Analytic conductor: \(6.51579\)
Root analytic conductor: \(2.55260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 816,\ (\ :1/2),\ -0.595 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.532220 - 1.05734i\)
\(L(\frac12)\) \(\approx\) \(0.532220 - 1.05734i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.39 + 1.03i)T \)
17 \( 1 + iT \)
good5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + 2.78iT - 7T^{2} \)
11 \( 1 - 3.50T + 11T^{2} \)
13 \( 1 - 5.74T + 13T^{2} \)
19 \( 1 + 5.56iT - 19T^{2} \)
23 \( 1 + 6.90T + 23T^{2} \)
29 \( 1 - 5.48iT - 29T^{2} \)
31 \( 1 + 3.50iT - 31T^{2} \)
37 \( 1 + 9.48T + 37T^{2} \)
41 \( 1 + 9.48iT - 41T^{2} \)
43 \( 1 - 7.00iT - 43T^{2} \)
47 \( 1 + 9.69T + 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 5.56T + 59T^{2} \)
61 \( 1 - 5.48T + 61T^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 - 4.93T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 11.7iT - 79T^{2} \)
83 \( 1 - 2.68T + 83T^{2} \)
89 \( 1 - 13.2iT - 89T^{2} \)
97 \( 1 + 9.48T + 97T^{2} \)
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

−10.08336884748023582845113417292, −8.942676242563880124600703853208, −8.298648989046437011668802874931, −7.13061935902118706874922578346, −6.56284307855279596928128548411, −5.56200246949769537348902662556, −4.55111039628679824909607350772, −3.70856134607764352851930437506, −1.62904722832477333854510973877, −0.71817579033110758031650731239, 1.67515416124514251740800634598, 3.38563934610217004218045967455, 4.02014955098730802438258691718, 5.46360449404046756738057323400, 6.23324853202053654982946749233, 6.60529865377650685951639441745, 8.163891476092807786478684320075, 8.905992895592701035579239781235, 9.902171516188693408298214275375, 10.50839665585413847055553702465

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