Properties

Label 2-170-17.16-c1-0-2
Degree $2$
Conductor $170$
Sign $0.970 - 0.242i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + i·5-s + 2i·7-s + 8-s + 3·9-s + i·10-s − 4i·11-s − 2·13-s + 2i·14-s + 16-s + (−1 − 4i)17-s + 3·18-s − 8·19-s + i·20-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.447i·5-s + 0.755i·7-s + 0.353·8-s + 9-s + 0.316i·10-s − 1.20i·11-s − 0.554·13-s + 0.534i·14-s + 0.250·16-s + (−0.242 − 0.970i)17-s + 0.707·18-s − 1.83·19-s + 0.223i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68391 + 0.207299i\)
\(L(\frac12)\) \(\approx\) \(1.68391 + 0.207299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - iT \)
17 \( 1 + (1 + 4i)T \)
good3 \( 1 - 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 10iT - 71T^{2} \)
73 \( 1 - 12iT - 73T^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.94407642036537484328674743374, −11.85433668855291833341100897616, −11.05543305765230238128605601836, −9.945609295567579133963362296170, −8.719628844099429559617781278284, −7.37919842515154954497070386172, −6.34769481191182450507979128760, −5.21583820190124830450662575045, −3.81144439505337770209116126257, −2.35577095351504871124232407459, 1.96873713218844423806568882524, 4.16309584125763120004185199517, 4.66481553244488061897850650772, 6.45168467505020539170819963411, 7.26835572617531933661978272572, 8.538087315103129226148980569814, 10.11639100699523557853309705068, 10.55622503560228451325784433348, 12.19336794839424565471569783485, 12.71991391950053845994315374925

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