↑

Properties

Label 2-170-85.63-c1-0-2
Degree $2$
Conductor $170$
Sign $0.197 + 0.980i$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
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 + (−1.83 − 0.364i)3-s + (0.707 + 0.707i)4-s + (−0.482 + 2.18i)5-s + (1.55 + 1.03i)6-s + (2.66 − 3.99i)7-s + (−0.382 − 0.923i)8-s + (0.456 + 0.189i)9-s + (1.28 − 1.83i)10-s + (1.22 − 1.83i)11-s + (−1.03 − 1.55i)12-s − 3.00i·13-s + (−3.99 + 2.66i)14-s + (1.68 − 3.82i)15-s + i·16-s + (−1.36 − 3.89i)17-s + ⋯
L(s)  = 1  + (−0.653 − 0.270i)2-s + (−1.05 − 0.210i)3-s + (0.353 + 0.353i)4-s + (−0.215 + 0.976i)5-s + (0.634 + 0.423i)6-s + (1.00 − 1.50i)7-s + (−0.135 − 0.326i)8-s + (0.152 + 0.0630i)9-s + (0.405 − 0.579i)10-s + (0.369 − 0.552i)11-s + (−0.299 − 0.448i)12-s − 0.833i·13-s + (−1.06 + 0.712i)14-s + (0.433 − 0.988i)15-s + 0.250i·16-s + (−0.330 − 0.943i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.197 + 0.980i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.197 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461073 - 0.377553i\)
\(L(\frac12)\) \(\approx\) \(0.461073 - 0.377553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 0.382i)T \)
5 \( 1 + (0.482 - 2.18i)T \)
17 \( 1 + (1.36 + 3.89i)T \)
good3 \( 1 + (1.83 + 0.364i)T + (2.77 + 1.14i)T^{2} \)
7 \( 1 + (-2.66 + 3.99i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (-1.22 + 1.83i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + 3.00iT - 13T^{2} \)
19 \( 1 + (-3.34 + 1.38i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.0960 - 0.482i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (-9.29 - 1.84i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (1.30 + 1.95i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (0.315 - 1.58i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (8.33 - 1.65i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (0.0317 - 0.0131i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 5.32T + 47T^{2} \)
53 \( 1 + (3.99 - 9.64i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.770 + 1.86i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.560 + 2.82i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (0.858 + 0.858i)T + 67iT^{2} \)
71 \( 1 + (-11.7 + 7.82i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-6.16 - 9.22i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (-4.06 - 2.71i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (6.45 + 2.67i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (2.81 + 2.81i)T + 89iT^{2} \)
97 \( 1 + (-9.98 - 14.9i)T + (-37.1 + 89.6i)T^{2} \)
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

−12.03555425481828496734356339620, −11.24916447572683996154292695097, −10.85168213159179668670517913631, −9.950155757136568573798476692370, −8.242234683503882294062410750044, −7.28188344024015160320872998495, −6.49567149899421873359503981137, −4.91488796085511302991995295936, −3.25280779091591047442128308077, −0.842725548220672789745127835008, 1.75680058959866750560343686172, 4.67679733257278819002261055304, 5.42728992508126684559307047426, 6.49750769873542174574348221237, 8.198577797065904574192271856690, 8.763094988926631407536880178056, 9.879354703734283941295638161046, 11.23559658882782561247576950309, 11.88444589346030711925355190730, 12.40358273756764996239192789041

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