↑

Properties

Label 2-1148-287.163-c1-0-10
Degree $2$
Conductor $1148$
Sign $0.381 - 0.924i$
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  + (1.85 − 1.07i)3-s + (−0.972 + 1.68i)5-s + (1.16 + 2.37i)7-s + (0.804 − 1.39i)9-s + (−3.48 + 2.01i)11-s − 1.38i·13-s + 4.17i·15-s + (−2.82 + 1.63i)17-s + (4.09 + 2.36i)19-s + (4.71 + 3.16i)21-s + (0.698 − 1.20i)23-s + (0.610 + 1.05i)25-s + 2.98i·27-s + 3.17i·29-s + (3.80 + 6.59i)31-s + ⋯
L(s)  = 1  + (1.07 − 0.619i)3-s + (−0.434 + 0.752i)5-s + (0.440 + 0.897i)7-s + (0.268 − 0.464i)9-s + (−1.04 + 0.606i)11-s − 0.384i·13-s + 1.07i·15-s + (−0.685 + 0.396i)17-s + (0.939 + 0.542i)19-s + (1.02 + 0.690i)21-s + (0.145 − 0.252i)23-s + (0.122 + 0.211i)25-s + 0.574i·27-s + 0.589i·29-s + (0.683 + 1.18i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.381 - 0.924i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.381 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.925860787\)
\(L(\frac12)\) \(\approx\) \(1.925860787\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.16 - 2.37i)T \)
41 \( 1 + (0.869 + 6.34i)T \)
good3 \( 1 + (-1.85 + 1.07i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.972 - 1.68i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.48 - 2.01i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.38iT - 13T^{2} \)
17 \( 1 + (2.82 - 1.63i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.09 - 2.36i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.698 + 1.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.17iT - 29T^{2} \)
31 \( 1 + (-3.80 - 6.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.38 + 2.39i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 3.50T + 43T^{2} \)
47 \( 1 + (7.73 + 4.46i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.48 + 2.00i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.14 - 3.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.617 - 1.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.30 - 4.21i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.37iT - 71T^{2} \)
73 \( 1 + (-6.61 - 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.50 + 1.44i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.06T + 83T^{2} \)
89 \( 1 + (-6.56 - 3.78i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.8iT - 97T^{2} \)
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

−9.943096379511135415810619595264, −8.776493808769172655921141783824, −8.365678539803829325163084020994, −7.47422370832663249934474828075, −7.04199840382209790770399666025, −5.72046761892779276785519127223, −4.83126170023446191545106148821, −3.35753675839433024451001800983, −2.69720740232978466781048154453, −1.79472952289267000092346166666, 0.73106111055828087278921592518, 2.48457139183137018323861871132, 3.44826330069312336508063271212, 4.45503297088253840934572409094, 4.91529197003938768059373272457, 6.32659762358325940390165312523, 7.64799381415775119428332445351, 8.000175801924436924278210962486, 8.822758336995560822016743569187, 9.532439000519755754282436887125

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