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Properties

Label 2-170-85.22-c1-0-1
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: $\chi_{170} (107, \cdot )$
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

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

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