Properties

Label 2-1148-41.32-c1-0-20
Degree $2$
Conductor $1148$
Sign $-0.783 + 0.621i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
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 − 1.65i)3-s − 2.95i·5-s + (−0.707 + 0.707i)7-s − 2.45i·9-s + (0.583 − 0.583i)11-s + (−0.304 + 0.304i)13-s + (−4.88 − 4.88i)15-s + (−4.59 − 4.59i)17-s + (−2.24 − 2.24i)19-s + 2.33i·21-s − 2.62·23-s − 3.72·25-s + (0.893 + 0.893i)27-s + (3.12 − 3.12i)29-s + 3.25·31-s + ⋯
L(s)  = 1  + (0.953 − 0.953i)3-s − 1.32i·5-s + (−0.267 + 0.267i)7-s − 0.819i·9-s + (0.175 − 0.175i)11-s + (−0.0843 + 0.0843i)13-s + (−1.26 − 1.26i)15-s + (−1.11 − 1.11i)17-s + (−0.514 − 0.514i)19-s + 0.509i·21-s − 0.546·23-s − 0.745·25-s + (0.171 + 0.171i)27-s + (0.580 − 0.580i)29-s + 0.584·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.783 + 0.621i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.870070267\)
\(L(\frac12)\) \(\approx\) \(1.870070267\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (6.24 - 1.41i)T \)
good3 \( 1 + (-1.65 + 1.65i)T - 3iT^{2} \)
5 \( 1 + 2.95iT - 5T^{2} \)
11 \( 1 + (-0.583 + 0.583i)T - 11iT^{2} \)
13 \( 1 + (0.304 - 0.304i)T - 13iT^{2} \)
17 \( 1 + (4.59 + 4.59i)T + 17iT^{2} \)
19 \( 1 + (2.24 + 2.24i)T + 19iT^{2} \)
23 \( 1 + 2.62T + 23T^{2} \)
29 \( 1 + (-3.12 + 3.12i)T - 29iT^{2} \)
31 \( 1 - 3.25T + 31T^{2} \)
37 \( 1 - 9.55T + 37T^{2} \)
43 \( 1 - 4.05iT - 43T^{2} \)
47 \( 1 + (7.44 + 7.44i)T + 47iT^{2} \)
53 \( 1 + (-0.577 + 0.577i)T - 53iT^{2} \)
59 \( 1 - 5.94T + 59T^{2} \)
61 \( 1 + 5.32iT - 61T^{2} \)
67 \( 1 + (-6.10 - 6.10i)T + 67iT^{2} \)
71 \( 1 + (7.75 - 7.75i)T - 71iT^{2} \)
73 \( 1 - 5.39iT - 73T^{2} \)
79 \( 1 + (-3.82 + 3.82i)T - 79iT^{2} \)
83 \( 1 - 4.27T + 83T^{2} \)
89 \( 1 + (-1.40 + 1.40i)T - 89iT^{2} \)
97 \( 1 + (-5.46 - 5.46i)T + 97iT^{2} \)
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

−9.222507364332531209254588212721, −8.521180835985238033136714509334, −8.141705884036513645820980055137, −7.03775131119282690027139540720, −6.33119068415534897637188242820, −5.06513907128038743789220526053, −4.29601716805904282846950532697, −2.86227405261891288363898728395, −2.00653623305084135358882638259, −0.70158030636500439279624008071, 2.14229150926251209393787920606, 3.08542284286396475075170397981, 3.84708533321525448780821721901, 4.59315434113661938464964780884, 6.19826716319602008695424409098, 6.68535373621291214390263452489, 7.82519915545490943594821339232, 8.540559704663345102852697862400, 9.407788274300175532907813646136, 10.25941128428107528501166877526

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