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Properties

Label 2-170-85.27-c1-0-2
Degree $2$
Conductor $170$
Sign $0.0845 + 0.996i$
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  + (−0.923 + 0.382i)2-s + (−2.97 + 0.592i)3-s + (0.707 − 0.707i)4-s + (−0.552 + 2.16i)5-s + (2.52 − 1.68i)6-s + (−0.562 − 0.842i)7-s + (−0.382 + 0.923i)8-s + (5.75 − 2.38i)9-s + (−0.318 − 2.21i)10-s + (−2.75 − 4.11i)11-s + (−1.68 + 2.52i)12-s − 6.20i·13-s + (0.842 + 0.562i)14-s + (0.362 − 6.78i)15-s − i·16-s + (1.65 + 3.77i)17-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (−1.72 + 0.342i)3-s + (0.353 − 0.353i)4-s + (−0.247 + 0.968i)5-s + (1.03 − 0.689i)6-s + (−0.212 − 0.318i)7-s + (−0.135 + 0.326i)8-s + (1.91 − 0.794i)9-s + (−0.100 − 0.699i)10-s + (−0.829 − 1.24i)11-s + (−0.487 + 0.729i)12-s − 1.72i·13-s + (0.225 + 0.150i)14-s + (0.0936 − 1.75i)15-s − 0.250i·16-s + (0.402 + 0.915i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0845 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0845 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.0845 + 0.996i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.0845 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162885 - 0.149647i\)
\(L(\frac12)\) \(\approx\) \(0.162885 - 0.149647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (0.552 - 2.16i)T \)
17 \( 1 + (-1.65 - 3.77i)T \)
good3 \( 1 + (2.97 - 0.592i)T + (2.77 - 1.14i)T^{2} \)
7 \( 1 + (0.562 + 0.842i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (2.75 + 4.11i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + 6.20iT - 13T^{2} \)
19 \( 1 + (-0.903 - 0.374i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.746 + 3.75i)T + (-21.2 - 8.80i)T^{2} \)
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 \( 1 + (0.885 + 4.45i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (3.09 + 0.616i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (8.74 + 3.62i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 2.54T + 47T^{2} \)
53 \( 1 + (-0.354 - 0.855i)T + (-37.4 + 37.4i)T^{2} \)
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 \( 1 + (-6.09 + 9.12i)T + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (3.24 - 2.17i)T + (30.2 - 72.9i)T^{2} \)
83 \( 1 + (-1.59 + 0.660i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.57 - 5.57i)T - 89iT^{2} \)
97 \( 1 + (0.560 - 0.838i)T + (-37.1 - 89.6i)T^{2} \)
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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

−12.24513288011219046825799039034, −11.07085251786475976064550905920, −10.58860050451199488704045737439, −10.13222640242758106179876946982, −8.238031553578606178620382218051, −7.16843231013813929612586284824, −6.02679061224405739475303463113, −5.42546270875633745242094724687, −3.42945443657008299556198700674, −0.31361725754650123145731523498, 1.65157512166080292537390218045, 4.52023491839951822825618243403, 5.42393130572403060499382562347, 6.81911175529585952249219156226, 7.63841306030666369046197215338, 9.272504186459666500726807925084, 9.972139090103360546169268782502, 11.37961440580905202336284403032, 11.82175914360678028043999345546, 12.55980724388048448711149973725

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