Properties

Label 2-170-85.78-c1-0-7
Degree $2$
Conductor $170$
Sign $-0.129 + 0.991i$
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  + (0.382 + 0.923i)2-s + (−0.592 − 2.97i)3-s + (−0.707 + 0.707i)4-s + (−1.33 − 1.79i)5-s + (2.52 − 1.68i)6-s + (−0.842 + 0.562i)7-s + (−0.923 − 0.382i)8-s + (−5.75 + 2.38i)9-s + (1.14 − 1.92i)10-s + (−2.75 − 4.11i)11-s + (2.52 + 1.68i)12-s + 6.20·13-s + (−0.842 − 0.562i)14-s + (−4.54 + 5.05i)15-s i·16-s + (3.77 − 1.65i)17-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (−0.342 − 1.72i)3-s + (−0.353 + 0.353i)4-s + (−0.599 − 0.800i)5-s + (1.03 − 0.689i)6-s + (−0.318 + 0.212i)7-s + (−0.326 − 0.135i)8-s + (−1.91 + 0.794i)9-s + (0.360 − 0.608i)10-s + (−0.829 − 1.24i)11-s + (0.729 + 0.487i)12-s + 1.72·13-s + (−0.225 − 0.150i)14-s + (−1.17 + 1.30i)15-s − 0.250i·16-s + (0.915 − 0.402i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.129 + 0.991i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.129 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.602127 - 0.686160i\)
\(L(\frac12)\) \(\approx\) \(0.602127 - 0.686160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 - 0.923i)T \)
5 \( 1 + (1.33 + 1.79i)T \)
17 \( 1 + (-3.77 + 1.65i)T \)
good3 \( 1 + (0.592 + 2.97i)T + (-2.77 + 1.14i)T^{2} \)
7 \( 1 + (0.842 - 0.562i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (2.75 + 4.11i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 - 6.20T + 13T^{2} \)
19 \( 1 + (0.903 + 0.374i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-3.75 - 0.746i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (-5.41 + 1.07i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + (0.672 - 1.00i)T + (-11.8 - 28.6i)T^{2} \)
37 \( 1 + (4.45 - 0.885i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (3.09 + 0.616i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (-3.62 + 8.74i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 2.54iT - 47T^{2} \)
53 \( 1 + (0.855 - 0.354i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.04 - 2.52i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (0.741 - 3.72i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-3.21 - 3.21i)T + 67iT^{2} \)
71 \( 1 + (4.85 + 3.24i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (-9.12 - 6.09i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-3.24 + 2.17i)T + (30.2 - 72.9i)T^{2} \)
83 \( 1 + (-0.660 - 1.59i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-5.57 + 5.57i)T - 89iT^{2} \)
97 \( 1 + (-0.838 - 0.560i)T + (37.1 + 89.6i)T^{2} \)
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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

−12.68424804552651510223005022680, −11.88998588559810884458122653053, −10.93636412592036813668272476572, −8.701554843938112567363079028791, −8.281455831757908250535805946860, −7.24751816798033958131897132252, −6.10700406554324325851658951304, −5.36519704749145689844500709152, −3.28615094052908092542078910715, −0.873015611119790841878674409764, 3.14690868272015241109747072371, 3.97250167868051499217163468949, 5.06157079662500673152592185609, 6.40811561086114100376006681699, 8.167194755058369357786621481755, 9.475548533182043560258583931246, 10.45875394608190122872153167320, 10.70835166527577517092624994275, 11.72676210536784989509177202164, 12.86717149982035772922988245954

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