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Properties

Label 2-170-85.58-c1-0-3
Degree $2$
Conductor $170$
Sign $0.807 + 0.589i$
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 + (−2.87 + 0.572i)3-s + (−0.707 + 0.707i)4-s + (2.22 + 0.258i)5-s + (1.63 + 2.44i)6-s + (−0.222 − 0.332i)7-s + (0.923 + 0.382i)8-s + (5.18 − 2.14i)9-s + (−0.611 − 2.15i)10-s + (3.38 − 2.25i)11-s + (1.63 − 2.44i)12-s + 1.42·13-s + (−0.222 + 0.332i)14-s + (−6.54 + 0.528i)15-s − i·16-s + (2.04 − 3.58i)17-s + ⋯
L(s)  = 1  + (−0.270 − 0.653i)2-s + (−1.66 + 0.330i)3-s + (−0.353 + 0.353i)4-s + (0.993 + 0.115i)5-s + (0.665 + 0.996i)6-s + (−0.0840 − 0.125i)7-s + (0.326 + 0.135i)8-s + (1.72 − 0.716i)9-s + (−0.193 − 0.680i)10-s + (1.01 − 0.681i)11-s + (0.470 − 0.704i)12-s + 0.395·13-s + (−0.0594 + 0.0889i)14-s + (−1.68 + 0.136i)15-s − 0.250i·16-s + (0.495 − 0.868i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.807 + 0.589i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.807 + 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.697609 - 0.227624i\)
\(L(\frac12)\) \(\approx\) \(0.697609 - 0.227624i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 - 0.258i)T \)
17 \( 1 + (-2.04 + 3.58i)T \)
good3 \( 1 + (2.87 - 0.572i)T + (2.77 - 1.14i)T^{2} \)
7 \( 1 + (0.222 + 0.332i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (-3.38 + 2.25i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 - 1.42T + 13T^{2} \)
19 \( 1 + (-3.96 - 1.64i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.20 - 6.07i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (-0.790 - 3.97i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (5.97 + 3.99i)T + (11.8 + 28.6i)T^{2} \)
37 \( 1 + (0.940 + 4.72i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (1.63 - 8.20i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (-4.46 + 10.7i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 7.15iT - 47T^{2} \)
53 \( 1 + (-4.41 + 1.83i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (1.95 + 4.72i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (10.8 + 2.14i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (0.229 + 0.229i)T + 67iT^{2} \)
71 \( 1 + (-2.55 + 3.82i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (7.33 - 10.9i)T + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (0.586 + 0.877i)T + (-30.2 + 72.9i)T^{2} \)
83 \( 1 + (-0.476 - 1.14i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (9.95 - 9.95i)T - 89iT^{2} \)
97 \( 1 + (-1.65 + 2.47i)T + (-37.1 - 89.6i)T^{2} \)
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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

−12.29119967381253573349519729528, −11.50992598050640935552076127101, −10.84649557698492470299920609475, −9.836467909894714332490536017086, −9.181600610694542143743276807810, −7.20821826094736075131810625103, −5.98004393833558623850060073121, −5.26022760503588820712896985882, −3.63690710878544291267796458671, −1.22450948689079608086563627299, 1.37498895190816731506824821885, 4.55991112447871095395162876120, 5.72117597044645239861899123744, 6.32756822070427830507436370676, 7.23911092825605489346334328599, 8.897549603197991823583594170810, 9.985124219385426389401240966984, 10.75739130520255831997806176401, 12.00765080312608177876342027496, 12.68934953077687548332419032449

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