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Properties

Label 2-170-85.3-c1-0-5
Degree $2$
Conductor $170$
Sign $0.468 + 0.883i$
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 + (−0.572 − 0.856i)3-s + (−0.707 − 0.707i)4-s + (−1.01 − 1.99i)5-s + (1.01 − 0.201i)6-s + (0.326 + 1.64i)7-s + (0.923 − 0.382i)8-s + (0.741 − 1.79i)9-s + (2.22 − 0.180i)10-s + (−1.12 − 5.67i)11-s + (−0.201 + 1.01i)12-s − 4.62i·13-s + (−1.64 − 0.326i)14-s + (−1.12 + 2.01i)15-s + i·16-s + (1.53 + 3.82i)17-s + ⋯
L(s)  = 1  + (−0.270 + 0.653i)2-s + (−0.330 − 0.494i)3-s + (−0.353 − 0.353i)4-s + (−0.455 − 0.890i)5-s + (0.412 − 0.0820i)6-s + (0.123 + 0.621i)7-s + (0.326 − 0.135i)8-s + (0.247 − 0.596i)9-s + (0.704 − 0.0569i)10-s + (−0.340 − 1.71i)11-s + (−0.0580 + 0.291i)12-s − 1.28i·13-s + (−0.439 − 0.0873i)14-s + (−0.289 + 0.519i)15-s + 0.250i·16-s + (0.372 + 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.468 + 0.883i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.468 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.650271 - 0.391304i\)
\(L(\frac12)\) \(\approx\) \(0.650271 - 0.391304i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.01 + 1.99i)T \)
17 \( 1 + (-1.53 - 3.82i)T \)
good3 \( 1 + (0.572 + 0.856i)T + (-1.14 + 2.77i)T^{2} \)
7 \( 1 + (-0.326 - 1.64i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (1.12 + 5.67i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + 4.62iT - 13T^{2} \)
19 \( 1 + (-0.932 - 2.25i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.78 - 1.85i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (3.70 + 5.55i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (0.914 - 4.59i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (1.36 - 0.910i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-6.23 + 9.32i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (-4.08 - 9.87i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 2.54T + 47T^{2} \)
53 \( 1 + (4.25 + 1.76i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-9.68 - 4.00i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.66 - 2.45i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (-1.91 - 1.91i)T + 67iT^{2} \)
71 \( 1 + (8.05 + 1.60i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (-1.06 + 5.37i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-4.50 + 0.895i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (-2.34 + 5.66i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-10.2 - 10.2i)T + 89iT^{2} \)
97 \( 1 + (-1.86 + 9.36i)T + (-89.6 - 37.1i)T^{2} \)
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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

−12.70684246741565889608111064360, −11.76737448728864085351385172329, −10.63560904753087065611246683208, −9.214231813904211314241156014727, −8.346478758793743579796474940216, −7.61847501104227162411614311352, −5.93720431301323819224275767471, −5.51234058850133149448728665231, −3.61812135968867637017666140564, −0.856732825726418730187163520028, 2.29432478665832295993648182379, 4.04721981901276965879047370342, 4.89381897101316575106211144534, 7.07382059373495467467564900884, 7.51604233738283509986531188717, 9.321427231747969633343203170219, 10.10828444742866105902027148679, 10.95169722479373139467271577920, 11.61633694430361029832313583577, 12.75009632658995656066451471553

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