Properties

Label 2-884-884.847-c0-0-0
Degree $2$
Conductor $884$
Sign $-0.225 - 0.974i$
Analytic cond. $0.441173$
Root an. cond. $0.664208$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.991 + 0.130i)2-s + (0.965 − 0.258i)4-s + (−0.389 + 1.95i)5-s + (−0.923 + 0.382i)8-s + (0.793 − 0.608i)9-s + (0.130 − 1.99i)10-s + (−0.608 + 0.793i)13-s + (0.866 − 0.5i)16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)18-s + (0.130 + 1.99i)20-s + (−2.75 − 1.14i)25-s + (0.499 − 0.866i)26-s + (−0.391 + 0.793i)29-s + (−0.793 + 0.608i)32-s + ⋯
L(s)  = 1  + (−0.991 + 0.130i)2-s + (0.965 − 0.258i)4-s + (−0.389 + 1.95i)5-s + (−0.923 + 0.382i)8-s + (0.793 − 0.608i)9-s + (0.130 − 1.99i)10-s + (−0.608 + 0.793i)13-s + (0.866 − 0.5i)16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)18-s + (0.130 + 1.99i)20-s + (−2.75 − 1.14i)25-s + (0.499 − 0.866i)26-s + (−0.391 + 0.793i)29-s + (−0.793 + 0.608i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 884 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(884\)    =    \(2^{2} \cdot 13 \cdot 17\)
Sign: $-0.225 - 0.974i$
Analytic conductor: \(0.441173\)
Root analytic conductor: \(0.664208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{884} (847, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 884,\ (\ :0),\ -0.225 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5857823166\)
\(L(\frac12)\) \(\approx\) \(0.5857823166\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.991 - 0.130i)T \)
13 \( 1 + (0.608 - 0.793i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.793 + 0.608i)T^{2} \)
5 \( 1 + (0.389 - 1.95i)T + (-0.923 - 0.382i)T^{2} \)
7 \( 1 + (-0.793 - 0.608i)T^{2} \)
11 \( 1 + (0.991 + 0.130i)T^{2} \)
19 \( 1 + (0.258 - 0.965i)T^{2} \)
23 \( 1 + (0.130 - 0.991i)T^{2} \)
29 \( 1 + (0.391 - 0.793i)T + (-0.608 - 0.793i)T^{2} \)
31 \( 1 + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (1.34 + 0.665i)T + (0.608 + 0.793i)T^{2} \)
41 \( 1 + (-0.423 - 0.483i)T + (-0.130 + 0.991i)T^{2} \)
43 \( 1 + (0.258 - 0.965i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.241 - 0.0999i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.965 + 0.258i)T^{2} \)
61 \( 1 + (-0.583 - 1.18i)T + (-0.608 + 0.793i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.991 + 0.130i)T^{2} \)
73 \( 1 + (-1.85 - 0.369i)T + (0.923 + 0.382i)T^{2} \)
79 \( 1 + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.732 + 0.835i)T + (-0.130 - 0.991i)T^{2} \)
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

−10.39333208880432701455724181762, −9.935463307612917473923728303998, −9.024937515153767750643776516216, −7.81571070173819460599840516564, −7.14354167485048618751378991780, −6.72457544236381875194786867444, −5.80032478628908633980869789083, −3.96079925221905127490250940483, −3.06038450425419013327317212404, −1.88728834406980217021795888065, 0.821521010873783412990451230843, 2.04856665946721987106615956922, 3.69889879025483692405536080482, 4.90597390432879235815447412161, 5.55081518930677564797399960030, 7.08832580168609819470907062592, 7.909079264322388646897797401691, 8.301753851981022385144775649601, 9.379941233825180728003275627410, 9.790002113711813747773500477350

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