↑

Properties

Label 2-1148-41.18-c1-0-17
Degree $2$
Conductor $1148$
Sign $-0.623 + 0.781i$
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  + 1.14·3-s + (−1.88 + 1.37i)5-s + (0.309 − 0.951i)7-s − 1.69·9-s + (−1.48 − 1.07i)11-s + (−0.348 − 1.07i)13-s + (−2.15 + 1.56i)15-s + (−1.87 − 1.35i)17-s + (0.684 − 2.10i)19-s + (0.353 − 1.08i)21-s + (−1.08 − 3.32i)23-s + (0.138 − 0.424i)25-s − 5.36·27-s + (6.21 − 4.51i)29-s + (−7.82 − 5.68i)31-s + ⋯
L(s)  = 1  + 0.659·3-s + (−0.844 + 0.613i)5-s + (0.116 − 0.359i)7-s − 0.564·9-s + (−0.446 − 0.324i)11-s + (−0.0966 − 0.297i)13-s + (−0.557 + 0.404i)15-s + (−0.453 − 0.329i)17-s + (0.156 − 0.482i)19-s + (0.0770 − 0.237i)21-s + (−0.225 − 0.693i)23-s + (0.0276 − 0.0849i)25-s − 1.03·27-s + (1.15 − 0.838i)29-s + (−1.40 − 1.02i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.623 + 0.781i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.623 + 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6465198054\)
\(L(\frac12)\) \(\approx\) \(0.6465198054\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.74 - 2.83i)T \)
good3 \( 1 - 1.14T + 3T^{2} \)
5 \( 1 + (1.88 - 1.37i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (1.48 + 1.07i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.348 + 1.07i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.87 + 1.35i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.684 + 2.10i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.08 + 3.32i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-6.21 + 4.51i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (7.82 + 5.68i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (4.87 - 3.54i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (2.98 + 9.19i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.394 + 1.21i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.49 - 6.89i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.0268 + 0.0826i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.19 + 6.75i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-12.4 + 9.06i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-8.95 - 6.50i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 1.47T + 83T^{2} \)
89 \( 1 + (-0.511 + 1.57i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.26 + 4.54i)T + (29.9 - 92.2i)T^{2} \)
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

−9.421071437320459234207080133241, −8.529481048351723452681535229754, −7.901156512012220166980607113989, −7.24445283759371941000711858034, −6.26864650876242678261650494674, −5.13907491965744014602644373305, −4.02856363651409347243760761541, −3.17916020768407774516965069179, −2.35126184982598047351047996791, −0.24390476868447243546522975766, 1.73074524685397677000937015820, 2.96653686612920758769516610145, 3.89532319928646619795901118840, 4.88885244484375998477994315604, 5.74090589048971906429613169222, 6.98408975083185779541098164655, 7.85726999665836877899469521415, 8.486997145143831926911614918998, 9.012072870215943885311864583670, 9.944369855013407025907210528883

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