↑

Properties

Label 2-170-85.9-c1-0-6
Degree $2$
Conductor $170$
Sign $0.330 + 0.943i$
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 + (0.771 − 1.86i)3-s + 1.00i·4-s + (1.71 + 1.43i)5-s + (−1.86 + 0.771i)6-s + (2.47 − 1.02i)7-s + (0.707 − 0.707i)8-s + (−0.750 − 0.750i)9-s + (−0.203 − 2.22i)10-s + (1.21 − 0.502i)11-s + (1.86 + 0.771i)12-s − 6.94·13-s + (−2.47 − 1.02i)14-s + (3.98 − 2.09i)15-s − 1.00·16-s + (−0.815 − 4.04i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.445 − 1.07i)3-s + 0.500i·4-s + (0.768 + 0.639i)5-s + (−0.760 + 0.314i)6-s + (0.935 − 0.387i)7-s + (0.250 − 0.250i)8-s + (−0.250 − 0.250i)9-s + (−0.0642 − 0.704i)10-s + (0.366 − 0.151i)11-s + (0.537 + 0.222i)12-s − 1.92·13-s + (−0.661 − 0.273i)14-s + (1.03 − 0.541i)15-s − 0.250·16-s + (−0.197 − 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.330 + 0.943i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.330 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.967161 - 0.686306i\)
\(L(\frac12)\) \(\approx\) \(0.967161 - 0.686306i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 1.43i)T \)
17 \( 1 + (0.815 + 4.04i)T \)
good3 \( 1 + (-0.771 + 1.86i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-2.47 + 1.02i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.21 + 0.502i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 6.94T + 13T^{2} \)
19 \( 1 + (3.81 - 3.81i)T - 19iT^{2} \)
23 \( 1 + (-0.333 - 0.804i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-0.562 + 1.35i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-5.75 - 2.38i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (2.67 - 6.46i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.84 + 6.87i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (4.16 - 4.16i)T - 43iT^{2} \)
47 \( 1 + 5.46T + 47T^{2} \)
53 \( 1 + (-6.13 - 6.13i)T + 53iT^{2} \)
59 \( 1 + (4.90 + 4.90i)T + 59iT^{2} \)
61 \( 1 + (-2.72 - 6.58i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 6.64iT - 67T^{2} \)
71 \( 1 + (-9.18 - 3.80i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (7.32 + 3.03i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-0.160 + 0.0663i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-5.79 - 5.79i)T + 83iT^{2} \)
89 \( 1 - 1.81iT - 89T^{2} \)
97 \( 1 + (5.24 + 2.17i)T + (68.5 + 68.5i)T^{2} \)
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

−12.45809146008770567759832539791, −11.69265456241320834965973962565, −10.47261414422156905496145975815, −9.690824009798060851682898989232, −8.346716941333088012647831686476, −7.43825440054807380168050185815, −6.67756614815712457117135170671, −4.82341202610927134772713917983, −2.71790468533518336649029596128, −1.69841039311855017896615566094, 2.17174423409130158042715552032, 4.50831792400683289411464139208, 5.14112948581364831766167776044, 6.67731704182081579029701849445, 8.208941726410515291668475632213, 8.967839739201094643862772121874, 9.778872722018457849175230272464, 10.50642923181043238546597055849, 11.91277417695650731436518965886, 13.04705302387611851939163105194

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