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Properties

Label 2-170-85.28-c1-0-8
Degree $2$
Conductor $170$
Sign $0.362 + 0.931i$
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 + (1.04 − 1.56i)3-s + (0.707 − 0.707i)4-s + (−1.24 − 1.85i)5-s + (0.366 − 1.84i)6-s + (−0.635 + 3.19i)7-s + (0.382 − 0.923i)8-s + (−0.199 − 0.481i)9-s + (−1.85 − 1.24i)10-s + (−0.447 − 0.0890i)11-s + (−0.366 − 1.84i)12-s − 0.527·13-s + (0.635 + 3.19i)14-s + (−4.19 − 0.00145i)15-s − i·16-s + (3.42 + 2.29i)17-s + ⋯
L(s)  = 1  + (0.653 − 0.270i)2-s + (0.601 − 0.900i)3-s + (0.353 − 0.353i)4-s + (−0.555 − 0.831i)5-s + (0.149 − 0.751i)6-s + (−0.240 + 1.20i)7-s + (0.135 − 0.326i)8-s + (−0.0664 − 0.160i)9-s + (−0.587 − 0.393i)10-s + (−0.134 − 0.0268i)11-s + (−0.105 − 0.531i)12-s − 0.146·13-s + (0.169 + 0.854i)14-s + (−1.08 − 0.000375i)15-s − 0.250i·16-s + (0.831 + 0.555i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.362 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.362 + 0.931i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.362 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43929 - 0.984215i\)
\(L(\frac12)\) \(\approx\) \(1.43929 - 0.984215i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.24 + 1.85i)T \)
17 \( 1 + (-3.42 - 2.29i)T \)
good3 \( 1 + (-1.04 + 1.56i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (0.635 - 3.19i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (0.447 + 0.0890i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + 0.527T + 13T^{2} \)
19 \( 1 + (1.16 - 2.80i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-6.25 + 4.17i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (5.65 + 3.78i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (3.44 - 0.684i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (6.88 + 4.60i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (6.26 - 4.18i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-1.48 - 0.617i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 12.7iT - 47T^{2} \)
53 \( 1 + (1.73 + 4.19i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.43 - 1.42i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (5.84 + 8.74i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (3.29 + 3.29i)T + 67iT^{2} \)
71 \( 1 + (2.03 + 10.2i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-1.47 - 7.39i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (0.0264 - 0.132i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (-9.21 + 3.81i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-5.04 + 5.04i)T - 89iT^{2} \)
97 \( 1 + (0.676 + 3.40i)T + (-89.6 + 37.1i)T^{2} \)
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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

−12.56481021919835778827962119676, −12.17274957446552795780739401687, −10.87422854946889700727597197758, −9.327210520645775841578492022130, −8.393399915363804688279307232757, −7.48190041659197419594043347429, −6.03667852093274418988575838782, −4.90608334846087562498033454172, −3.26098270982238581764934765754, −1.81962496845686275991932681942, 3.19439243137422874165400989397, 3.79792684193795399651479383596, 5.08089815389505941842923345732, 6.91841670512284814844054403705, 7.43869309952630424433939206899, 8.944151390108600370292949250018, 10.15592253427654505766803458968, 10.84242384216041072108991111479, 11.94719575481785704523655827516, 13.29224059223580584347983272338

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