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Properties

Label 2-170-85.62-c1-0-7
Degree $2$
Conductor $170$
Sign $-0.745 + 0.666i$
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 + (−0.247 − 0.165i)3-s + (−0.707 + 0.707i)4-s + (−0.0264 − 2.23i)5-s + (−0.0581 + 0.292i)6-s + (−3.22 − 0.640i)7-s + (0.923 + 0.382i)8-s + (−1.11 − 2.68i)9-s + (−2.05 + 0.880i)10-s + (2.53 + 0.504i)11-s + (0.292 − 0.0581i)12-s − 1.91i·13-s + (0.640 + 3.22i)14-s + (−0.363 + 0.558i)15-s − i·16-s + (−2.98 − 2.84i)17-s + ⋯
L(s)  = 1  + (−0.270 − 0.653i)2-s + (−0.143 − 0.0956i)3-s + (−0.353 + 0.353i)4-s + (−0.0118 − 0.999i)5-s + (−0.0237 + 0.119i)6-s + (−1.21 − 0.242i)7-s + (0.326 + 0.135i)8-s + (−0.371 − 0.896i)9-s + (−0.650 + 0.278i)10-s + (0.765 + 0.152i)11-s + (0.0844 − 0.0167i)12-s − 0.531i·13-s + (0.171 + 0.860i)14-s + (−0.0939 + 0.144i)15-s − 0.250i·16-s + (−0.723 − 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.745 + 0.666i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.745 + 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.260023 - 0.681566i\)
\(L(\frac12)\) \(\approx\) \(0.260023 - 0.681566i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0264 + 2.23i)T \)
17 \( 1 + (2.98 + 2.84i)T \)
good3 \( 1 + (0.247 + 0.165i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (3.22 + 0.640i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (-2.53 - 0.504i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + 1.91iT - 13T^{2} \)
19 \( 1 + (-0.0675 + 0.163i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.23 - 3.34i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (-7.00 - 4.68i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (-7.76 + 1.54i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (-2.41 + 3.61i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-1.17 + 0.784i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-2.61 + 6.31i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 7.66T + 47T^{2} \)
53 \( 1 + (-2.61 + 1.08i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (13.0 - 5.38i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.752 + 1.12i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (5.06 - 5.06i)T - 67iT^{2} \)
71 \( 1 + (-2.15 - 10.8i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (9.52 - 1.89i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (-2.20 + 11.1i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (3.01 + 7.27i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-8.71 + 8.71i)T - 89iT^{2} \)
97 \( 1 + (-5.49 + 1.09i)T + (89.6 - 37.1i)T^{2} \)
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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

−12.29858691073280837207081142475, −11.66243401223423502628592238641, −10.25621264066042094846325968792, −9.281922889767585794588184712213, −8.781176787747678239582968895916, −7.14576553562709801899111403388, −5.95301347616057159644837952367, −4.38066544962385264313399463058, −3.09266292675971862985012836728, −0.77739216827680088315103205422, 2.75066696941234811510775145156, 4.40995308170022395934738430701, 6.27608552190089870451809682241, 6.49990113244004701186478842188, 7.951769644494460116111059908227, 9.095948088221408374859617444181, 10.11847487402107028585696769048, 10.95261586913366904717070629453, 12.11357875546594171778859809141, 13.47357284335218289892376000115

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