↑

Properties

Label 2-1148-41.16-c1-0-14
Degree $2$
Conductor $1148$
Sign $-0.982 + 0.186i$
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  − 2.16·3-s + (−0.636 − 0.462i)5-s + (−0.309 − 0.951i)7-s + 1.69·9-s + (2.63 − 1.91i)11-s + (0.271 − 0.835i)13-s + (1.37 + 1.00i)15-s + (−0.414 + 0.301i)17-s + (2.08 + 6.42i)19-s + (0.669 + 2.05i)21-s + (0.977 − 3.00i)23-s + (−1.35 − 4.16i)25-s + 2.83·27-s + (−3.96 − 2.87i)29-s + (−3.01 + 2.19i)31-s + ⋯
L(s)  = 1  − 1.25·3-s + (−0.284 − 0.206i)5-s + (−0.116 − 0.359i)7-s + 0.563·9-s + (0.793 − 0.576i)11-s + (0.0753 − 0.231i)13-s + (0.355 + 0.258i)15-s + (−0.100 + 0.0730i)17-s + (0.478 + 1.47i)19-s + (0.146 + 0.449i)21-s + (0.203 − 0.627i)23-s + (−0.270 − 0.833i)25-s + 0.545·27-s + (−0.735 − 0.534i)29-s + (−0.541 + 0.393i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.982 + 0.186i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.982 + 0.186i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2570297402\)
\(L(\frac12)\) \(\approx\) \(0.2570297402\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.178 - 6.40i)T \)
good3 \( 1 + 2.16T + 3T^{2} \)
5 \( 1 + (0.636 + 0.462i)T + (1.54 + 4.75i)T^{2} \)
11 \( 1 + (-2.63 + 1.91i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.271 + 0.835i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.414 - 0.301i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.08 - 6.42i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.977 + 3.00i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.96 + 2.87i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.01 - 2.19i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (6.62 + 4.81i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (1.58 - 4.88i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (0.934 - 2.87i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.37 + 6.81i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.31 + 10.2i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.22 + 6.85i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (5.33 + 3.87i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (6.51 - 4.73i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + 2.70T + 73T^{2} \)
79 \( 1 + 9.49T + 79T^{2} \)
83 \( 1 + 6.85T + 83T^{2} \)
89 \( 1 + (-3.19 - 9.83i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (15.0 + 10.9i)T + (29.9 + 92.2i)T^{2} \)
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

−9.597388650540154348842133768240, −8.515268540532988377535514993990, −7.76297948582213787081900425591, −6.62511962316820004003504231829, −6.08143144421588753286477014907, −5.24733645335180807750555328767, −4.24552166002798358937716546366, −3.33416141731348439997257215359, −1.45859477836407274379910086965, −0.14139073465590234777909578756, 1.52950808724547071932175999255, 3.10140844444218919082871664226, 4.27760200587348238590183471543, 5.22464228884186669926114374423, 5.86800809164366182789134399113, 7.02030205187938887660370189407, 7.21080885464252766521164099036, 8.808702395151854983821383267110, 9.313406870811134037764308856790, 10.35702256936310917327887566684

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