↑

Properties

Label 2-930-155.129-c1-0-27
Degree $2$
Conductor $930$
Sign $0.149 + 0.988i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  + i·2-s + (−0.866 − 0.5i)3-s − 4-s + (0.594 − 2.15i)5-s + (0.5 − 0.866i)6-s + (−0.759 − 0.438i)7-s − i·8-s + (0.499 + 0.866i)9-s + (2.15 + 0.594i)10-s + (1.33 + 2.31i)11-s + (0.866 + 0.5i)12-s + (0.519 − 0.299i)13-s + (0.438 − 0.759i)14-s + (−1.59 + 1.56i)15-s + 16-s + (−1.86 − 1.07i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.499 − 0.288i)3-s − 0.5·4-s + (0.265 − 0.964i)5-s + (0.204 − 0.353i)6-s + (−0.287 − 0.165i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.681 + 0.187i)10-s + (0.402 + 0.697i)11-s + (0.249 + 0.144i)12-s + (0.144 − 0.0831i)13-s + (0.117 − 0.202i)14-s + (−0.411 + 0.405i)15-s + 0.250·16-s + (−0.451 − 0.260i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.149 + 0.988i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.149 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.683863 - 0.588466i\)
\(L(\frac12)\) \(\approx\) \(0.683863 - 0.588466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.594 + 2.15i)T \)
31 \( 1 + (-5.42 - 1.25i)T \)
good7 \( 1 + (0.759 + 0.438i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.33 - 2.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.519 + 0.299i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.86 + 1.07i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.673 + 1.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.93iT - 23T^{2} \)
29 \( 1 + 1.51T + 29T^{2} \)
37 \( 1 + (9.54 + 5.50i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.64 + 6.31i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.33 + 3.65i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.66iT - 47T^{2} \)
53 \( 1 + (-0.455 + 0.263i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.97 + 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 2.40T + 61T^{2} \)
67 \( 1 + (3.85 - 2.22i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.57 - 13.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.34 - 0.774i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.08 - 5.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-14.7 + 8.53i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.54T + 89T^{2} \)
97 \( 1 - 2.08iT - 97T^{2} \)
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

−9.808729074775381940682953479712, −8.847828730483151715874547136906, −8.300383586566917824270316282899, −7.03358584243322844000420045969, −6.62919089844447462155802355412, −5.46453543955122001867897060829, −4.84719143418307604231550171407, −3.85954595439568854646001557734, −1.98483337946052097076702264742, −0.46392655091913745019161847757, 1.55491706266004461880914724442, 3.01707811616802594402847086514, 3.68400538324414638214195870992, 4.93149478578961289145648529820, 6.02018125404031370665630270808, 6.57488528520090018292615761717, 7.78340798729456608584748893167, 8.850974400186334249490752954435, 9.727023817003692454066981130192, 10.26504008154481067923686585145

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