Properties

Label 2-1176-7.4-c1-0-2
Degree $2$
Conductor $1176$
Sign $-0.968 - 0.250i$
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  + (−0.5 + 0.866i)3-s + (1 + 1.73i)5-s + (−0.499 − 0.866i)9-s − 6·13-s − 1.99·15-s + (−1 + 1.73i)17-s + (2 + 3.46i)19-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s + 0.999·27-s − 10·29-s + (−4 + 6.92i)31-s + (−3 − 5.19i)37-s + (3 − 5.19i)39-s + 2·41-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.447 + 0.774i)5-s + (−0.166 − 0.288i)9-s − 1.66·13-s − 0.516·15-s + (−0.242 + 0.420i)17-s + (0.458 + 0.794i)19-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s + 0.192·27-s − 1.85·29-s + (−0.718 + 1.24i)31-s + (−0.493 − 0.854i)37-s + (0.480 − 0.832i)39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.968 - 0.250i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7844261298\)
\(L(\frac12)\) \(\approx\) \(0.7844261298\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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

−10.04354299434950630379329126594, −9.622151790428445623978252087047, −8.664369333523431711358116766626, −7.41120207817121338067024244356, −6.97711500539021089001584579725, −5.75816789866569191500459359061, −5.22243255562075892538003599034, −4.01831156198972822280915653800, −3.03351787981709487428635675897, −1.90139602237741707520076894849, 0.32724813576113310732363766896, 1.81548823975343686464451720630, 2.83141404464371095700004839157, 4.43563142560415855989057335507, 5.17964723337731099353374177999, 5.86820060596529701074009142113, 7.21087800099089310048626629160, 7.40427154454011012039317843162, 8.773848756108570776951634174150, 9.296642119705502506989701178662

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