↑

Properties

Label 2-1148-41.16-c1-0-3
Degree $2$
Conductor $1148$
Sign $-0.460 - 0.887i$
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  − 0.842·3-s + (2.54 + 1.85i)5-s + (−0.309 − 0.951i)7-s − 2.29·9-s + (0.696 − 0.505i)11-s + (−1.28 + 3.95i)13-s + (−2.14 − 1.56i)15-s + (−3.99 + 2.90i)17-s + (0.379 + 1.16i)19-s + (0.260 + 0.801i)21-s + (−0.845 + 2.60i)23-s + (1.52 + 4.68i)25-s + 4.45·27-s + (2.90 + 2.10i)29-s + (−1.65 + 1.20i)31-s + ⋯
L(s)  = 1  − 0.486·3-s + (1.14 + 0.828i)5-s + (−0.116 − 0.359i)7-s − 0.763·9-s + (0.209 − 0.152i)11-s + (−0.356 + 1.09i)13-s + (−0.554 − 0.402i)15-s + (−0.968 + 0.703i)17-s + (0.0870 + 0.268i)19-s + (0.0568 + 0.174i)21-s + (−0.176 + 0.542i)23-s + (0.304 + 0.937i)25-s + 0.857·27-s + (0.538 + 0.391i)29-s + (−0.297 + 0.216i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.460 - 0.887i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.460 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.060491119\)
\(L(\frac12)\) \(\approx\) \(1.060491119\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (6.18 - 1.66i)T \)
good3 \( 1 + 0.842T + 3T^{2} \)
5 \( 1 + (-2.54 - 1.85i)T + (1.54 + 4.75i)T^{2} \)
11 \( 1 + (-0.696 + 0.505i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.28 - 3.95i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.99 - 2.90i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.379 - 1.16i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.845 - 2.60i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.90 - 2.10i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.65 - 1.20i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.03 + 2.20i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-1.12 + 3.45i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-0.0692 + 0.213i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.91 - 1.39i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.28 - 13.1i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.89 - 8.89i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (2.73 + 1.98i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-2.37 + 1.72i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 - 2.33T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (-1.36 - 4.20i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.78 - 2.02i)T + (29.9 + 92.2i)T^{2} \)
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

−10.31476225409601736115525722767, −9.253093921206452230707482956548, −8.672920649037442575728797753304, −7.31481727089100540335110431596, −6.54226897195377768337431304488, −6.05268988815299354788237762868, −5.11431054519375593745888190242, −3.92462521081533278644332017522, −2.71978808423705905720192563354, −1.69992274626188283693613413278, 0.46213872313479440246531023140, 2.04045106776831237748689607473, 3.03180363038339759054450908756, 4.72893770986417455007814671304, 5.26201574791581924108597082443, 6.03271962723297714618309439264, 6.77234030173558685780531780343, 8.104705042884521247225362935684, 8.823267826568664459956487057440, 9.508205577121904790620257781222

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