↑

Properties

Label 2-1176-8.5-c1-0-21
Degree $2$
Conductor $1176$
Sign $-0.726 - 0.687i$
Analytic cond. $9.39040$
Root an. cond. $3.06437$
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  + (−0.353 + 1.36i)2-s + i·3-s + (−1.75 − 0.967i)4-s − 0.677i·5-s + (−1.36 − 0.353i)6-s + (1.94 − 2.05i)8-s − 9-s + (0.927 + 0.239i)10-s + 1.67i·11-s + (0.967 − 1.75i)12-s + 1.28i·13-s + 0.677·15-s + (2.12 + 3.38i)16-s + 6.36·17-s + (0.353 − 1.36i)18-s − 2.55i·19-s + ⋯
L(s)  = 1  + (−0.249 + 0.968i)2-s + 0.577i·3-s + (−0.875 − 0.483i)4-s − 0.302i·5-s + (−0.559 − 0.144i)6-s + (0.687 − 0.726i)8-s − 0.333·9-s + (0.293 + 0.0756i)10-s + 0.503i·11-s + (0.279 − 0.505i)12-s + 0.355i·13-s + 0.174·15-s + (0.531 + 0.846i)16-s + 1.54·17-s + (0.0832 − 0.322i)18-s − 0.585i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.726 - 0.687i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.726 - 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.198433294\)
\(L(\frac12)\) \(\approx\) \(1.198433294\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.353 - 1.36i)T \)
3 \( 1 - iT \)
7 \( 1 \)
good5 \( 1 + 0.677iT - 5T^{2} \)
11 \( 1 - 1.67iT - 11T^{2} \)
13 \( 1 - 1.28iT - 13T^{2} \)
17 \( 1 - 6.36T + 17T^{2} \)
19 \( 1 + 2.55iT - 19T^{2} \)
23 \( 1 + 0.255T + 23T^{2} \)
29 \( 1 - 6.27iT - 29T^{2} \)
31 \( 1 + 4.28T + 31T^{2} \)
37 \( 1 - 6.49iT - 37T^{2} \)
41 \( 1 + 6.43T + 41T^{2} \)
43 \( 1 - 5.48iT - 43T^{2} \)
47 \( 1 - 9.46T + 47T^{2} \)
53 \( 1 - 5.67iT - 53T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 - 15.2iT - 61T^{2} \)
67 \( 1 - 15.7iT - 67T^{2} \)
71 \( 1 + 5.48T + 71T^{2} \)
73 \( 1 - 2.86T + 73T^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + 7.63iT - 83T^{2} \)
89 \( 1 - 6.80T + 89T^{2} \)
97 \( 1 + 0.477T + 97T^{2} \)
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

−9.982311405719972600910347197315, −9.076918152685356013062304276698, −8.594059998589453518337370968109, −7.53954847827431288959276326665, −6.89868838569505661799336154211, −5.78295649378656043678345634263, −5.06798423039091446350988734288, −4.31815485731476592542603498027, −3.16338950347352443313597429646, −1.25445151231524135037595649461, 0.66254226534169940124304381502, 1.93774902825428083885418273904, 3.08028734998574170878697989437, 3.80276961577537153861046095092, 5.21044711639607271806394193618, 5.96503618725259311379554252311, 7.26753931807026089272676095138, 7.923769214911342374822370622411, 8.673183888865787602572940412564, 9.591155278346392311773493716358

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