Properties

Label 2-170-17.16-c1-0-3
Degree $2$
Conductor $170$
Sign $0.242 + 0.970i$
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 i·3-s + 4-s i·5-s + i·6-s − 8-s + 2·9-s + i·10-s − 6i·11-s i·12-s − 5·13-s − 15-s + 16-s + (4 − i)17-s − 2·18-s + 3·19-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577i·3-s + 0.5·4-s − 0.447i·5-s + 0.408i·6-s − 0.353·8-s + 0.666·9-s + 0.316i·10-s − 1.80i·11-s − 0.288i·12-s − 1.38·13-s − 0.258·15-s + 0.250·16-s + (0.970 − 0.242i)17-s − 0.471·18-s + 0.688·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.242 + 0.970i$
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: \((2,\ 170,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.661850 - 0.516757i\)
\(L(\frac12)\) \(\approx\) \(0.661850 - 0.516757i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 + i)T \)
good3 \( 1 + iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 6iT - 11T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 - iT - 29T^{2} \)
31 \( 1 - 7iT - 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 12iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 7T + 47T^{2} \)
53 \( 1 + T + 53T^{2} \)
59 \( 1 + 9T + 59T^{2} \)
61 \( 1 + 3iT - 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 + 9iT - 71T^{2} \)
73 \( 1 - 7iT - 73T^{2} \)
79 \( 1 + 8iT - 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 - iT - 97T^{2} \)
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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

−12.36265902060199657020855747812, −11.70416152727181662473690338497, −10.41184485445080752737826829294, −9.517224951693734740414588796622, −8.351184073823879274480511207488, −7.55489761825516621496059546104, −6.41214605013478270568276589895, −5.06925577053071545335203107003, −3.05268001774041829351668466912, −1.08657951342704377928838709233, 2.19185027504633142046958447704, 4.03515190005163281835807438076, 5.38363727405958443972670767649, 7.25067159768810349849370774076, 7.47260858679348041521214588364, 9.407551739869589008132607484608, 9.849399957671859471824036832083, 10.60233511927826921436502384283, 12.00723022924032505724382508335, 12.61603016955695792603578429928

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