↑

Properties

Label 2-170-17.8-c1-0-5
Degree $2$
Conductor $170$
Sign $0.0465 + 0.998i$
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  + (0.707 − 0.707i)2-s + (0.865 − 2.08i)3-s − 1.00i·4-s + (0.923 + 0.382i)5-s + (−0.865 − 2.08i)6-s + (−2.01 + 0.834i)7-s + (−0.707 − 0.707i)8-s + (−1.49 − 1.49i)9-s + (0.923 − 0.382i)10-s + (1.18 + 2.86i)11-s + (−2.08 − 0.865i)12-s + 1.91i·13-s + (−0.834 + 2.01i)14-s + (1.59 − 1.59i)15-s − 1.00·16-s + (−3.85 + 1.46i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.499 − 1.20i)3-s − 0.500i·4-s + (0.413 + 0.171i)5-s + (−0.353 − 0.853i)6-s + (−0.761 + 0.315i)7-s + (−0.250 − 0.250i)8-s + (−0.498 − 0.498i)9-s + (0.292 − 0.121i)10-s + (0.357 + 0.862i)11-s + (−0.603 − 0.249i)12-s + 0.531i·13-s + (−0.222 + 0.538i)14-s + (0.412 − 0.412i)15-s − 0.250·16-s + (−0.934 + 0.355i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0465 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0465 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.0465 + 0.998i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.0465 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19997 - 1.14540i\)
\(L(\frac12)\) \(\approx\) \(1.19997 - 1.14540i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (3.85 - 1.46i)T \)
good3 \( 1 + (-0.865 + 2.08i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (2.01 - 0.834i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.18 - 2.86i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.91iT - 13T^{2} \)
19 \( 1 + (-3.27 + 3.27i)T - 19iT^{2} \)
23 \( 1 + (2.44 + 5.89i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-6.49 - 2.68i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (1.66 - 4.00i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (3.12 - 7.53i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.375 - 0.155i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (5.56 + 5.56i)T + 43iT^{2} \)
47 \( 1 + 0.118iT - 47T^{2} \)
53 \( 1 + (-2.75 + 2.75i)T - 53iT^{2} \)
59 \( 1 + (6.89 + 6.89i)T + 59iT^{2} \)
61 \( 1 + (14.0 - 5.81i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 9.13T + 67T^{2} \)
71 \( 1 + (-2.33 + 5.63i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-1.53 - 0.636i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.76 - 9.09i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-5.90 + 5.90i)T - 83iT^{2} \)
89 \( 1 + 9.71iT - 89T^{2} \)
97 \( 1 + (-1.28 - 0.532i)T + (68.5 + 68.5i)T^{2} \)
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

−12.51470761727843121289458124261, −12.05222327231576552314567214044, −10.60287180299983052476417835380, −9.515461091429527892331640990240, −8.509545411308632992747898436312, −6.85877330460255275401885997548, −6.51803866090665914428863615430, −4.72466435261272975488876554735, −2.93769616930904709185463459481, −1.81238477827705778710355203900, 3.13656965323417773821674642630, 4.04100767146005173399762752418, 5.38133228791288033295916346518, 6.45859578104986559158498610418, 7.944316297096432000269513731035, 9.140979588956277637341851418820, 9.786468728020615976630102389053, 10.86970451847139799609221649992, 12.14835188978955798747257358473, 13.44776195931650457785253884689

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