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Properties

Label 2-170-85.82-c1-0-0
Degree $2$
Conductor $170$
Sign $0.985 - 0.168i$
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 + (−0.839 − 1.25i)3-s + (0.707 + 0.707i)4-s + (0.291 + 2.21i)5-s + (0.294 + 1.48i)6-s + (0.823 + 4.13i)7-s + (−0.382 − 0.923i)8-s + (0.273 − 0.660i)9-s + (0.578 − 2.15i)10-s + (4.92 − 0.980i)11-s + (0.294 − 1.48i)12-s − 1.01·13-s + (0.823 − 4.13i)14-s + (2.54 − 2.22i)15-s + i·16-s + (−1.64 − 3.78i)17-s + ⋯
L(s)  = 1  + (−0.653 − 0.270i)2-s + (−0.484 − 0.725i)3-s + (0.353 + 0.353i)4-s + (0.130 + 0.991i)5-s + (0.120 + 0.605i)6-s + (0.311 + 1.56i)7-s + (−0.135 − 0.326i)8-s + (0.0912 − 0.220i)9-s + (0.183 − 0.682i)10-s + (1.48 − 0.295i)11-s + (0.0851 − 0.427i)12-s − 0.282·13-s + (0.219 − 1.10i)14-s + (0.656 − 0.575i)15-s + 0.250i·16-s + (−0.398 − 0.917i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.985 - 0.168i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.985 - 0.168i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.811574 + 0.0688518i\)
\(L(\frac12)\) \(\approx\) \(0.811574 + 0.0688518i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.291 - 2.21i)T \)
17 \( 1 + (1.64 + 3.78i)T \)
good3 \( 1 + (0.839 + 1.25i)T + (-1.14 + 2.77i)T^{2} \)
7 \( 1 + (-0.823 - 4.13i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-4.92 + 0.980i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + 1.01T + 13T^{2} \)
19 \( 1 + (-2.35 - 5.67i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-6.31 - 4.21i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (-0.994 + 0.664i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (1.71 + 0.341i)T + (28.6 + 11.8i)T^{2} \)
37 \( 1 + (0.956 - 0.639i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (3.25 + 2.17i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (-5.07 + 2.10i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 11.5iT - 47T^{2} \)
53 \( 1 + (1.77 - 4.29i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (10.1 + 4.18i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (0.916 - 1.37i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (4.09 - 4.09i)T - 67iT^{2} \)
71 \( 1 + (-1.92 + 9.69i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (-2.05 + 10.3i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-0.444 - 2.23i)T + (-72.9 + 30.2i)T^{2} \)
83 \( 1 + (-3.85 - 1.59i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (1.31 + 1.31i)T + 89iT^{2} \)
97 \( 1 + (-2.46 + 12.3i)T + (-89.6 - 37.1i)T^{2} \)
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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

−12.19369045618816264452790264772, −11.87887518205884153200957110141, −11.09937618451188265433912028121, −9.596094419930785674599256744381, −8.949619180317519746242523261916, −7.47552596995494101694564942279, −6.58359338001692158121523670965, −5.65649429304415577373353170127, −3.30476461587231888572702061495, −1.76628447338916120947022383380, 1.18947702387363472136192445169, 4.21422547796053842086347880907, 4.88943143638084343219291713206, 6.54500636744228021958827539631, 7.57203358430039672101177538816, 8.892237644475501802218473780246, 9.669404932251861848299104112726, 10.71159661685835975672933455781, 11.33988888046143233511263836184, 12.72647729736394773221024827252

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