↑

Properties

Label 2-930-155.27-c1-0-4
Degree $2$
Conductor $930$
Sign $0.0402 - 0.999i$
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  + (−0.453 + 0.891i)2-s + (−0.891 + 0.453i)3-s + (−0.587 − 0.809i)4-s + (−0.789 − 2.09i)5-s − 1.00i·6-s + (−4.01 − 0.635i)7-s + (0.987 − 0.156i)8-s + (0.587 − 0.809i)9-s + (2.22 + 0.245i)10-s + (−0.883 − 1.21i)11-s + (0.891 + 0.453i)12-s + (−4.09 + 2.08i)13-s + (2.38 − 3.28i)14-s + (1.65 + 1.50i)15-s + (−0.309 + 0.951i)16-s + (−0.206 + 0.0327i)17-s + ⋯
L(s)  = 1  + (−0.321 + 0.630i)2-s + (−0.514 + 0.262i)3-s + (−0.293 − 0.404i)4-s + (−0.353 − 0.935i)5-s − 0.408i·6-s + (−1.51 − 0.240i)7-s + (0.349 − 0.0553i)8-s + (0.195 − 0.269i)9-s + (0.702 + 0.0777i)10-s + (−0.266 − 0.366i)11-s + (0.257 + 0.131i)12-s + (−1.13 + 0.578i)13-s + (0.637 − 0.878i)14-s + (0.426 + 0.388i)15-s + (−0.0772 + 0.237i)16-s + (−0.0501 + 0.00794i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0402 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0402 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.0402 - 0.999i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.0402 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.373870 + 0.359126i\)
\(L(\frac12)\) \(\approx\) \(0.373870 + 0.359126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.453 - 0.891i)T \)
3 \( 1 + (0.891 - 0.453i)T \)
5 \( 1 + (0.789 + 2.09i)T \)
31 \( 1 + (-5.17 - 2.06i)T \)
good7 \( 1 + (4.01 + 0.635i)T + (6.65 + 2.16i)T^{2} \)
11 \( 1 + (0.883 + 1.21i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (4.09 - 2.08i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.206 - 0.0327i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-6.35 + 2.06i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.736 - 4.64i)T + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (-1.22 - 3.77i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (-0.473 + 0.473i)T - 37iT^{2} \)
41 \( 1 + (-2.38 - 7.32i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-5.57 + 10.9i)T + (-25.2 - 34.7i)T^{2} \)
47 \( 1 + (-6.23 + 3.17i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-1.76 - 11.1i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (8.64 + 2.80i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 - 8.30iT - 61T^{2} \)
67 \( 1 + (7.87 + 7.87i)T + 67iT^{2} \)
71 \( 1 + (-8.24 - 5.98i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-9.94 - 1.57i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (11.6 + 8.44i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.07 - 9.96i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (1.48 - 1.07i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (16.9 + 2.68i)T + (92.2 + 29.9i)T^{2} \)
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

−9.887386464325063423050561986538, −9.483471131925417133052357013595, −8.793279889498251736185389425360, −7.49552641774273415300383608393, −7.02531141056361842642891742847, −5.90105887919555720436547803928, −5.18179561104871509635045468874, −4.24290073752395751682471957888, −3.05786714146659581344407468910, −0.881054221632391013840580339499, 0.40276894954564467757884826660, 2.53247157128237010563519403844, 3.07835152423751907799438701700, 4.33212111023347200904756412484, 5.63682023182357486625562153178, 6.54558544876598888291653171658, 7.34003736895075081254112685172, 8.022347920066758282196024767478, 9.514720413741660839081072330072, 9.902400508444600016883816621429

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