Properties

Label 2-1176-168.59-c0-0-6
Degree $2$
Conductor $1176$
Sign $-0.294 + 0.955i$
Analytic cond. $0.586900$
Root an. cond. $0.766094$
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  + (0.866 − 0.5i)2-s + (0.130 − 0.991i)3-s + (0.499 − 0.866i)4-s + (−0.382 − 0.923i)6-s − 0.999i·8-s + (−0.965 − 0.258i)9-s + (1.22 + 0.707i)11-s + (−0.793 − 0.608i)12-s + (−0.5 − 0.866i)16-s + (0.382 − 0.662i)17-s + (−0.965 + 0.258i)18-s + (−1.60 + 0.923i)19-s + 1.41·22-s + (−0.991 − 0.130i)24-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.130 − 0.991i)3-s + (0.499 − 0.866i)4-s + (−0.382 − 0.923i)6-s − 0.999i·8-s + (−0.965 − 0.258i)9-s + (1.22 + 0.707i)11-s + (−0.793 − 0.608i)12-s + (−0.5 − 0.866i)16-s + (0.382 − 0.662i)17-s + (−0.965 + 0.258i)18-s + (−1.60 + 0.923i)19-s + 1.41·22-s + (−0.991 − 0.130i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.294 + 0.955i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ -0.294 + 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.769766599\)
\(L(\frac12)\) \(\approx\) \(1.769766599\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.130 + 0.991i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.84T + T^{2} \)
43 \( 1 + 1.41T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - 0.765T + T^{2} \)
89 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.84iT - T^{2} \)
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

−9.744202846385355430366826334366, −9.047003829998635012710287666176, −7.899619750505257566538310238750, −7.00430900569692073246960798974, −6.36706563556503414648159052194, −5.58343019021888808855797014833, −4.38321072744372134252689648740, −3.50710210725827676929358906334, −2.27683047824529867166977844547, −1.40469594582521019491252182490, 2.32379401628011227533442206774, 3.51865376714167412029723928370, 4.14665276207051570727330971917, 4.94424778159783348198474763519, 6.12905902649710301633953981380, 6.43600140648504717068467578711, 7.83348092127495797839870548432, 8.605394504170754823074743508465, 9.176872214931083523221858111914, 10.35261714350855680500324308637

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