↑

Properties

Label 2-1014-13.12-c3-0-74
Degree $2$
Conductor $1014$
Sign $-0.722 - 0.691i$
Analytic cond. $59.8279$
Root an. cond. $7.73485$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + 3·3-s − 4·4-s − 13.0i·5-s − 6i·6-s − 14.8i·7-s + 8i·8-s + 9·9-s − 26.0·10-s − 39.1i·11-s − 12·12-s − 29.6·14-s − 39.1i·15-s + 16·16-s − 8.45·17-s − 18i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 1.16i·5-s − 0.408i·6-s − 0.799i·7-s + 0.353i·8-s + 0.333·9-s − 0.824·10-s − 1.07i·11-s − 0.288·12-s − 0.565·14-s − 0.673i·15-s + 0.250·16-s − 0.120·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.722 - 0.691i$
Analytic conductor: \(59.8279\)
Root analytic conductor: \(7.73485\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :3/2),\ -0.722 - 0.691i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.624926577\)
\(L(\frac12)\) \(\approx\) \(1.624926577\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 - 3T \)
13 \( 1 \)
good5 \( 1 + 13.0iT - 125T^{2} \)
7 \( 1 + 14.8iT - 343T^{2} \)
11 \( 1 + 39.1iT - 1.33e3T^{2} \)
17 \( 1 + 8.45T + 4.91e3T^{2} \)
19 \( 1 + 33.2iT - 6.85e3T^{2} \)
23 \( 1 + 173.T + 1.21e4T^{2} \)
29 \( 1 - 142.T + 2.43e4T^{2} \)
31 \( 1 + 268. iT - 2.97e4T^{2} \)
37 \( 1 - 201. iT - 5.06e4T^{2} \)
41 \( 1 + 165. iT - 6.89e4T^{2} \)
43 \( 1 + 365.T + 7.95e4T^{2} \)
47 \( 1 - 175. iT - 1.03e5T^{2} \)
53 \( 1 - 711.T + 1.48e5T^{2} \)
59 \( 1 + 190. iT - 2.05e5T^{2} \)
61 \( 1 + 634.T + 2.26e5T^{2} \)
67 \( 1 + 432. iT - 3.00e5T^{2} \)
71 \( 1 - 801. iT - 3.57e5T^{2} \)
73 \( 1 - 171. iT - 3.89e5T^{2} \)
79 \( 1 - 604.T + 4.93e5T^{2} \)
83 \( 1 - 1.23e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.06e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.51e3iT - 9.12e5T^{2} \)
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

−9.050751741257788598426134199845, −8.365002819266072577936270004428, −7.79703507474711576739249250458, −6.46043518077017339446049248901, −5.35049226490437542072162796739, −4.34864834261335729326566182542, −3.72076732451880344995074379284, −2.49616060938282170612344508936, −1.23630060564818997326199404743, −0.38367963887912883955649867801, 1.86400496985825685166077770215, 2.82576286781550254425166269437, 3.87476335366365421769967019027, 4.95539499398671637676504951999, 6.05625992694435093131538885549, 6.80562162353170166400756346482, 7.49736270306085325260996969145, 8.356561036475757976485342769893, 9.117499999306623832085374506495, 10.11170200502143116074130603278

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