↑

Properties

Label 2-510-85.28-c1-0-0
Degree $2$
Conductor $510$
Sign $-0.989 + 0.142i$
Analytic cond. $4.07237$
Root an. cond. $2.01801$
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  + (−0.923 + 0.382i)2-s + (0.555 − 0.831i)3-s + (0.707 − 0.707i)4-s + (−1.07 + 1.96i)5-s + (−0.195 + 0.980i)6-s + (−0.251 + 1.26i)7-s + (−0.382 + 0.923i)8-s + (−0.382 − 0.923i)9-s + (0.241 − 2.22i)10-s + (−6.22 − 1.23i)11-s + (−0.195 − 0.980i)12-s − 0.969·13-s + (−0.251 − 1.26i)14-s + (1.03 + 1.98i)15-s − i·16-s + (−3.91 − 1.28i)17-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (0.320 − 0.480i)3-s + (0.353 − 0.353i)4-s + (−0.480 + 0.877i)5-s + (−0.0796 + 0.400i)6-s + (−0.0951 + 0.478i)7-s + (−0.135 + 0.326i)8-s + (−0.127 − 0.307i)9-s + (0.0765 − 0.702i)10-s + (−1.87 − 0.373i)11-s + (−0.0563 − 0.283i)12-s − 0.268·13-s + (−0.0673 − 0.338i)14-s + (0.266 + 0.511i)15-s − 0.250i·16-s + (−0.950 − 0.311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(510\)    =    \(2 \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.989 + 0.142i$
Analytic conductor: \(4.07237\)
Root analytic conductor: \(2.01801\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{510} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 510,\ (\ :1/2),\ -0.989 + 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00406001 - 0.0568935i\)
\(L(\frac12)\) \(\approx\) \(0.00406001 - 0.0568935i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 - 0.382i)T \)
3 \( 1 + (-0.555 + 0.831i)T \)
5 \( 1 + (1.07 - 1.96i)T \)
17 \( 1 + (3.91 + 1.28i)T \)
good7 \( 1 + (0.251 - 1.26i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (6.22 + 1.23i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + 0.969T + 13T^{2} \)
19 \( 1 + (-0.720 + 1.73i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.44 + 1.63i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (1.47 + 0.984i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (8.50 - 1.69i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (-6.78 - 4.53i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (8.11 - 5.42i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (3.41 + 1.41i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 3.12iT - 47T^{2} \)
53 \( 1 + (-4.27 - 10.3i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (7.80 - 3.23i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (8.24 + 12.3i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (6.33 + 6.33i)T + 67iT^{2} \)
71 \( 1 + (-1.19 - 5.99i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-1.92 - 9.68i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (0.614 - 3.08i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (-15.9 + 6.59i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (0.492 - 0.492i)T - 89iT^{2} \)
97 \( 1 + (-1.77 - 8.91i)T + (-89.6 + 37.1i)T^{2} \)
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

−11.08302692363162516536155574471, −10.59819401539888627789525173053, −9.480338524147640629262296926711, −8.523722988521338216446330866979, −7.74037655834101411293986426619, −7.08706308057252636217811614605, −6.10177716802527342615656330215, −4.94402493837215330205952405548, −3.08522484291457951915343621710, −2.34290177428959811573774581076, 0.03659613973925145654871300714, 2.07367790161958497179406677951, 3.47726927028781112651629514839, 4.59912789375683241600938532459, 5.52804001312830469178545075010, 7.28978794986944112572692882432, 7.82922426627525985567715324767, 8.747280095314621108127252961382, 9.502449506544650120705701692140, 10.45775001012187658814548959625

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