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Properties

Label 2-170-85.27-c1-0-4
Degree $2$
Conductor $170$
Sign $0.992 + 0.123i$
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 + (0.917 − 0.182i)3-s + (0.707 − 0.707i)4-s + (0.304 + 2.21i)5-s + (0.778 − 0.519i)6-s + (1.08 + 1.62i)7-s + (0.382 − 0.923i)8-s + (−1.96 + 0.812i)9-s + (1.12 + 1.93i)10-s + (−2.02 − 3.02i)11-s + (0.519 − 0.778i)12-s − 6.15i·13-s + (1.62 + 1.08i)14-s + (0.683 + 1.97i)15-s − i·16-s + (−4.05 − 0.742i)17-s + ⋯
L(s)  = 1  + (0.653 − 0.270i)2-s + (0.529 − 0.105i)3-s + (0.353 − 0.353i)4-s + (0.135 + 0.990i)5-s + (0.317 − 0.212i)6-s + (0.410 + 0.614i)7-s + (0.135 − 0.326i)8-s + (−0.654 + 0.270i)9-s + (0.356 + 0.610i)10-s + (−0.609 − 0.911i)11-s + (0.150 − 0.224i)12-s − 1.70i·13-s + (0.434 + 0.290i)14-s + (0.176 + 0.510i)15-s − 0.250i·16-s + (−0.983 − 0.180i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.992 + 0.123i$
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: \((2,\ 170,\ (\ :1/2),\ 0.992 + 0.123i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81788 - 0.113131i\)
\(L(\frac12)\) \(\approx\) \(1.81788 - 0.113131i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.304 - 2.21i)T \)
17 \( 1 + (4.05 + 0.742i)T \)
good3 \( 1 + (-0.917 + 0.182i)T + (2.77 - 1.14i)T^{2} \)
7 \( 1 + (-1.08 - 1.62i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (2.02 + 3.02i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + 6.15iT - 13T^{2} \)
19 \( 1 + (-2.19 - 0.909i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.700 - 3.52i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (-5.01 + 0.996i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + (0.311 - 0.466i)T + (-11.8 - 28.6i)T^{2} \)
37 \( 1 + (-0.893 - 4.49i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-4.93 - 0.981i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (-7.27 - 3.01i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 12.6T + 47T^{2} \)
53 \( 1 + (2.84 + 6.86i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-5.12 - 12.3i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-0.856 + 4.30i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (5.21 - 5.21i)T - 67iT^{2} \)
71 \( 1 + (-4.65 - 3.11i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (1.32 - 1.98i)T + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (-11.1 + 7.44i)T + (30.2 - 72.9i)T^{2} \)
83 \( 1 + (9.70 - 4.02i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (1.02 - 1.02i)T - 89iT^{2} \)
97 \( 1 + (-8.76 + 13.1i)T + (-37.1 - 89.6i)T^{2} \)
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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

−13.03795769577410198489324648806, −11.59090837425758374600599911565, −10.99842134397186532127612848167, −9.992756949564955110949084910403, −8.482946075784121314897019425853, −7.66655923562075620599379125439, −6.08769943663375428841958378023, −5.27381291342142129301315220210, −3.24120200417148325881738463818, −2.55553290351038896885197792268, 2.19417690837771306831372915070, 4.15055870446989176853235186057, 4.85478327301415412603448166137, 6.38934630843348527939646484148, 7.60980753396621608430768008620, 8.693958240480783530432932872686, 9.518794690483611223009605080858, 11.02027068257130820441137003590, 12.01781345831388306924125697926, 12.92129126203740277846862109436

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