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Properties

Label 2-170-85.19-c1-0-0
Degree $2$
Conductor $170$
Sign $-0.737 - 0.674i$
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  + (−0.707 + 0.707i)2-s + (0.961 + 2.32i)3-s − 1.00i·4-s + (−1.24 + 1.85i)5-s + (−2.32 − 0.961i)6-s + (−0.124 − 0.0515i)7-s + (0.707 + 0.707i)8-s + (−2.34 + 2.34i)9-s + (−0.434 − 2.19i)10-s + (−1.51 − 0.627i)11-s + (2.32 − 0.961i)12-s + 3.30·13-s + (0.124 − 0.0515i)14-s + (−5.50 − 1.09i)15-s − 1.00·16-s + (−2.59 − 3.20i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.555 + 1.33i)3-s − 0.500i·4-s + (−0.556 + 0.831i)5-s + (−0.947 − 0.392i)6-s + (−0.0470 − 0.0194i)7-s + (0.250 + 0.250i)8-s + (−0.780 + 0.780i)9-s + (−0.137 − 0.693i)10-s + (−0.456 − 0.189i)11-s + (0.669 − 0.277i)12-s + 0.917·13-s + (0.0332 − 0.0137i)14-s + (−1.42 − 0.283i)15-s − 0.250·16-s + (−0.629 − 0.777i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.737 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.737 - 0.674i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.737 - 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.341915 + 0.880562i\)
\(L(\frac12)\) \(\approx\) \(0.341915 + 0.880562i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.24 - 1.85i)T \)
17 \( 1 + (2.59 + 3.20i)T \)
good3 \( 1 + (-0.961 - 2.32i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (0.124 + 0.0515i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.51 + 0.627i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 3.30T + 13T^{2} \)
19 \( 1 + (-3.24 - 3.24i)T + 19iT^{2} \)
23 \( 1 + (1.86 - 4.49i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-2.75 - 6.65i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-6.99 + 2.89i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (1.21 + 2.92i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.66 + 4.02i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-1.56 - 1.56i)T + 43iT^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 + (-5.73 + 5.73i)T - 53iT^{2} \)
59 \( 1 + (0.746 - 0.746i)T - 59iT^{2} \)
61 \( 1 + (1.57 - 3.80i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 11.9iT - 67T^{2} \)
71 \( 1 + (12.0 - 5.00i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (14.0 - 5.82i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-6.14 - 2.54i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.434 + 0.434i)T - 83iT^{2} \)
89 \( 1 + 12.1iT - 89T^{2} \)
97 \( 1 + (0.683 - 0.283i)T + (68.5 - 68.5i)T^{2} \)
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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

−13.65143636619505629411471292743, −11.76910188584557532939956712267, −10.77625498329214504202389646311, −10.12707894533770611338068420912, −9.112189172793703038269469517188, −8.182385382735692770438589342322, −7.08413306891361760794811512859, −5.63927688908761594829682189003, −4.18133682516532922146350450101, −3.04590202189468044530751146274, 1.08597618463753994809133078377, 2.64589142962766547446805795150, 4.36235039020772086195839354085, 6.29765039192413510607123779106, 7.54640125828653079557473783180, 8.315914518905654026309735797773, 8.960224221467482709291565658189, 10.44990733406778381041229474882, 11.69637801267396043653045432245, 12.39119997520852772965621735412

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