Properties

Label 2-170-85.84-c1-0-5
Degree $2$
Conductor $170$
Sign $0.759 + 0.650i$
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  i·2-s + 3-s − 4-s + (2 + i)5-s i·6-s + 2·7-s + i·8-s − 2·9-s + (1 − 2i)10-s − 12-s i·13-s − 2i·14-s + (2 + i)15-s + 16-s + (1 − 4i)17-s + 2i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.5·4-s + (0.894 + 0.447i)5-s − 0.408i·6-s + 0.755·7-s + 0.353i·8-s − 0.666·9-s + (0.316 − 0.632i)10-s − 0.288·12-s − 0.277i·13-s − 0.534i·14-s + (0.516 + 0.258i)15-s + 0.250·16-s + (0.242 − 0.970i)17-s + 0.471i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.759 + 0.650i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.759 + 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35518 - 0.501313i\)
\(L(\frac12)\) \(\approx\) \(1.35518 - 0.501313i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (-2 - i)T \)
17 \( 1 + (-1 + 4i)T \)
good3 \( 1 - T + 3T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 9iT - 29T^{2} \)
31 \( 1 + 5iT - 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 + 5T + 59T^{2} \)
61 \( 1 + 5iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 5iT - 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 16iT - 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 5T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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

−12.73069282427223356333378913942, −11.49070460681623677730087814455, −10.73606964720735632063528440668, −9.673699245760630854857163727186, −8.777267495738734926545470281019, −7.73819026466125478177770246183, −6.12219164428167399966994400260, −4.87878126295597667599082294631, −3.16218883380341411967148272410, −2.01602626977355442406190385855, 2.09774505601165615279768778452, 4.14800988415631466917326004810, 5.48830190311268484103928206059, 6.40224127653083741685001868836, 8.051648952256609540644552125398, 8.553894901425455625758113621537, 9.595427966477503798189291150409, 10.71369463785094807460861887954, 12.09850529905754912727863072647, 13.20003916449535018258867736667

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