↑

Properties

Label 2-170-17.4-c1-0-0
Degree $2$
Conductor $170$
Sign $0.633 - 0.773i$
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 + (−1.20 + 1.20i)3-s − 4-s + (−0.707 + 0.707i)5-s + (1.20 + 1.20i)6-s + (1.69 + 1.69i)7-s + i·8-s + 0.113i·9-s + (0.707 + 0.707i)10-s + (3.82 + 3.82i)11-s + (1.20 − 1.20i)12-s − 2.28·13-s + (1.69 − 1.69i)14-s − 1.69i·15-s + 16-s + (−2.20 + 3.48i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.693 + 0.693i)3-s − 0.5·4-s + (−0.316 + 0.316i)5-s + (0.490 + 0.490i)6-s + (0.642 + 0.642i)7-s + 0.353i·8-s + 0.0377i·9-s + (0.223 + 0.223i)10-s + (1.15 + 1.15i)11-s + (0.346 − 0.346i)12-s − 0.633·13-s + (0.454 − 0.454i)14-s − 0.438i·15-s + 0.250·16-s + (−0.533 + 0.845i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.633 - 0.773i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.633 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.763984 + 0.362041i\)
\(L(\frac12)\) \(\approx\) \(0.763984 + 0.362041i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (2.20 - 3.48i)T \)
good3 \( 1 + (1.20 - 1.20i)T - 3iT^{2} \)
7 \( 1 + (-1.69 - 1.69i)T + 7iT^{2} \)
11 \( 1 + (-3.82 - 3.82i)T + 11iT^{2} \)
13 \( 1 + 2.28T + 13T^{2} \)
19 \( 1 + 3.51iT - 19T^{2} \)
23 \( 1 + (0.585 + 0.585i)T + 23iT^{2} \)
29 \( 1 + (-0.384 + 0.384i)T - 29iT^{2} \)
31 \( 1 + (-6.72 + 6.72i)T - 31iT^{2} \)
37 \( 1 + (3.12 - 3.12i)T - 37iT^{2} \)
41 \( 1 + (-4.39 - 4.39i)T + 41iT^{2} \)
43 \( 1 - 1.59iT - 43T^{2} \)
47 \( 1 - 4.11T + 47T^{2} \)
53 \( 1 - 1.11iT - 53T^{2} \)
59 \( 1 - 11.1iT - 59T^{2} \)
61 \( 1 + (9.01 + 9.01i)T + 61iT^{2} \)
67 \( 1 - 8.46T + 67T^{2} \)
71 \( 1 + (-9.13 + 9.13i)T - 71iT^{2} \)
73 \( 1 + (-8.74 + 8.74i)T - 73iT^{2} \)
79 \( 1 + (-11.6 - 11.6i)T + 79iT^{2} \)
83 \( 1 + 10.9iT - 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + (-10.5 + 10.5i)T - 97iT^{2} \)
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

−12.47218736241756186102478381450, −11.73875218588566111270840169516, −11.08045070093740634921712430047, −10.07588096045429278842822268621, −9.202935466393771007846784242610, −7.901066405237870468016177271746, −6.39634294777869207780356235919, −4.89776488161756380938423009154, −4.20789672873558533290541434201, −2.21131061375701842408846412428, 0.942079757485420219575953977218, 3.88877916364050574657788383359, 5.22384422439175403315096085846, 6.41359805067691878748755152721, 7.22797657311700088184966847159, 8.315311161120892504393127849067, 9.359132635359870081485321946973, 10.88229934011883394080114426968, 11.80982239143961299303944309398, 12.49742947166106852670127404835

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