Properties

Label 2-1152-96.11-c1-0-11
Degree $2$
Conductor $1152$
Sign $0.861 + 0.507i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
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  + (1.65 + 0.686i)5-s + (0.456 − 0.456i)7-s + (−2.79 − 1.15i)11-s + (−1.29 − 3.12i)13-s + 3.55·17-s + (5.61 − 2.32i)19-s + (4.10 − 4.10i)23-s + (−1.26 − 1.26i)25-s + (1.87 + 4.53i)29-s − 0.580i·31-s + (1.06 − 0.442i)35-s + (2.58 − 6.22i)37-s + (2.98 + 2.98i)41-s + (2.78 − 6.72i)43-s + 8.67i·47-s + ⋯
L(s)  = 1  + (0.740 + 0.306i)5-s + (0.172 − 0.172i)7-s + (−0.841 − 0.348i)11-s + (−0.359 − 0.867i)13-s + 0.862·17-s + (1.28 − 0.533i)19-s + (0.855 − 0.855i)23-s + (−0.252 − 0.252i)25-s + (0.348 + 0.841i)29-s − 0.104i·31-s + (0.180 − 0.0748i)35-s + (0.424 − 1.02i)37-s + (0.465 + 0.465i)41-s + (0.424 − 1.02i)43-s + 1.26i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.861 + 0.507i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.861 + 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.851869619\)
\(L(\frac12)\) \(\approx\) \(1.851869619\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (-1.65 - 0.686i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.456 + 0.456i)T - 7iT^{2} \)
11 \( 1 + (2.79 + 1.15i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.29 + 3.12i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 3.55T + 17T^{2} \)
19 \( 1 + (-5.61 + 2.32i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-4.10 + 4.10i)T - 23iT^{2} \)
29 \( 1 + (-1.87 - 4.53i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 0.580iT - 31T^{2} \)
37 \( 1 + (-2.58 + 6.22i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.98 - 2.98i)T + 41iT^{2} \)
43 \( 1 + (-2.78 + 6.72i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 8.67iT - 47T^{2} \)
53 \( 1 + (3.70 - 8.95i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.77 + 4.28i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (9.49 - 3.93i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (3.50 + 8.46i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-7.84 - 7.84i)T + 71iT^{2} \)
73 \( 1 + (-10.7 + 10.7i)T - 73iT^{2} \)
79 \( 1 - 5.27T + 79T^{2} \)
83 \( 1 + (-1.80 - 4.36i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-12.4 + 12.4i)T - 89iT^{2} \)
97 \( 1 - 9.99T + 97T^{2} \)
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

−9.732225552429595541674102868510, −9.076380987676144769567793393026, −7.86734425020647863909533904464, −7.45030983510006504576282594321, −6.24401915462440064462035944738, −5.48665265678561196870437543315, −4.75557001195935851138761064821, −3.21590721193818539286313688497, −2.54837704382073680732420390190, −0.919818538629635314271177550025, 1.34126042211011409169925412060, 2.45330581739027024958829081312, 3.63476668753347619839544312933, 5.02942672008161005189925025762, 5.39761017277139989837687416949, 6.48363971289059436791859566079, 7.51433952094441044896254623956, 8.112147853478171268529234621923, 9.401425379513398906899617936740, 9.633585045912876397311456802721

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