↑

Properties

Label 2-1148-7.4-c1-0-4
Degree $2$
Conductor $1148$
Sign $-0.677 - 0.735i$
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  + (−0.154 + 0.266i)3-s + (1.65 + 2.85i)5-s + (−2.57 − 0.626i)7-s + (1.45 + 2.51i)9-s + (−0.758 + 1.31i)11-s + 1.10·13-s − 1.01·15-s + (−0.803 + 1.39i)17-s + (−0.440 − 0.762i)19-s + (0.563 − 0.589i)21-s + (−0.341 − 0.590i)23-s + (−2.94 + 5.10i)25-s − 1.81·27-s − 1.88·29-s + (−2.35 + 4.08i)31-s + ⋯
L(s)  = 1  + (−0.0889 + 0.154i)3-s + (0.738 + 1.27i)5-s + (−0.971 − 0.236i)7-s + (0.484 + 0.838i)9-s + (−0.228 + 0.396i)11-s + 0.307·13-s − 0.262·15-s + (−0.194 + 0.337i)17-s + (−0.101 − 0.174i)19-s + (0.122 − 0.128i)21-s + (−0.0711 − 0.123i)23-s + (−0.589 + 1.02i)25-s − 0.350·27-s − 0.349·29-s + (−0.423 + 0.733i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.677 - 0.735i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.677 - 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.278202639\)
\(L(\frac12)\) \(\approx\) \(1.278202639\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.57 + 0.626i)T \)
41 \( 1 - T \)
good3 \( 1 + (0.154 - 0.266i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.65 - 2.85i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.758 - 1.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.10T + 13T^{2} \)
17 \( 1 + (0.803 - 1.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.440 + 0.762i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.341 + 0.590i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.88T + 29T^{2} \)
31 \( 1 + (2.35 - 4.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.61 - 2.78i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 9.57T + 43T^{2} \)
47 \( 1 + (2.39 + 4.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.816 + 1.41i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.505 - 0.874i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.190 - 0.329i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.24 - 7.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.90T + 71T^{2} \)
73 \( 1 + (0.782 - 1.35i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.20 - 7.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.55T + 83T^{2} \)
89 \( 1 + (6.96 + 12.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.601T + 97T^{2} \)
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

−10.15133087456943056793553257110, −9.640618152765495891980489980617, −8.489964209581913879339221357637, −7.35042452851637653321601450123, −6.78859033418708788114583616514, −6.06633866130609347165588165942, −5.04868026173609242274970101020, −3.83167340116480541903009540719, −2.86768456927419029556152925911, −1.88411129911344876440277706659, 0.54388646777992558038876241270, 1.81310453460918440499172963584, 3.24088097238156934293316608447, 4.27557220536999856360287292386, 5.39799545620293111749528064823, 6.05150457391075360884929173979, 6.80989700449168612838573971732, 7.979528626810629125043234912593, 8.949483465965308708737661712960, 9.416964824800727477711238361706

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