Properties

Label 2-1148-7.2-c1-0-8
Degree $2$
Conductor $1148$
Sign $-0.729 - 0.683i$
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  + (0.702 + 1.21i)3-s + (−1.80 + 3.13i)5-s + (0.606 + 2.57i)7-s + (0.514 − 0.890i)9-s + (2.82 + 4.89i)11-s + 6.54·13-s − 5.07·15-s + (−1.69 − 2.93i)17-s + (−2.59 + 4.50i)19-s + (−2.70 + 2.54i)21-s + (1.40 − 2.43i)23-s + (−4.03 − 6.99i)25-s + 5.65·27-s + 2.17·29-s + (−3.13 − 5.42i)31-s + ⋯
L(s)  = 1  + (0.405 + 0.702i)3-s + (−0.808 + 1.40i)5-s + (0.229 + 0.973i)7-s + (0.171 − 0.296i)9-s + (0.851 + 1.47i)11-s + 1.81·13-s − 1.31·15-s + (−0.411 − 0.712i)17-s + (−0.596 + 1.03i)19-s + (−0.590 + 0.555i)21-s + (0.292 − 0.507i)23-s + (−0.807 − 1.39i)25-s + 1.08·27-s + 0.404·29-s + (−0.562 − 0.973i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.729 - 0.683i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.729 - 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.812586552\)
\(L(\frac12)\) \(\approx\) \(1.812586552\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.606 - 2.57i)T \)
41 \( 1 + T \)
good3 \( 1 + (-0.702 - 1.21i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.80 - 3.13i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.82 - 4.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.54T + 13T^{2} \)
17 \( 1 + (1.69 + 2.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.59 - 4.50i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.40 + 2.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.17T + 29T^{2} \)
31 \( 1 + (3.13 + 5.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.320 - 0.555i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 4.44T + 43T^{2} \)
47 \( 1 + (3.34 - 5.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.47 + 4.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.51 - 9.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.01 + 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.29 + 9.16i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.70T + 71T^{2} \)
73 \( 1 + (0.211 + 0.366i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.11 + 12.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.06T + 83T^{2} \)
89 \( 1 + (-0.782 + 1.35i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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

−10.11031238609174309625609359272, −9.287450522722458416329245540828, −8.597580828187520901554003412662, −7.68903370872747973566543947773, −6.64365284099808334388228132228, −6.22911436054024288835581745397, −4.63202359502664161422996952277, −3.86717051959342903896354087978, −3.16676997423612026841348201103, −1.92137981746431637238217029653, 0.863916997437974828989215219197, 1.47929933960898215156272212102, 3.48363353446338195558549917913, 4.07786899324403801938686783605, 5.07011393510101185053519053959, 6.29690848235934109264243423618, 7.07164138633828104910705962565, 8.155419089989009429765231725499, 8.574665804515968752668636722412, 8.928209235441355575649760575260

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