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Properties

Label 2-170-85.22-c1-0-3
Degree $2$
Conductor $170$
Sign $0.733 - 0.680i$
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 + (3.34 + 0.664i)3-s + (−0.707 − 0.707i)4-s + (−0.732 − 2.11i)5-s + (−1.89 + 2.83i)6-s + (−0.616 + 0.923i)7-s + (0.923 − 0.382i)8-s + (7.94 + 3.29i)9-s + (2.23 + 0.131i)10-s + (−0.833 − 0.556i)11-s + (−1.89 − 2.83i)12-s − 4.43·13-s + (−0.616 − 0.923i)14-s + (−1.04 − 7.54i)15-s + i·16-s + (−3.76 + 1.67i)17-s + ⋯
L(s)  = 1  + (−0.270 + 0.653i)2-s + (1.92 + 0.383i)3-s + (−0.353 − 0.353i)4-s + (−0.327 − 0.944i)5-s + (−0.772 + 1.15i)6-s + (−0.233 + 0.348i)7-s + (0.326 − 0.135i)8-s + (2.64 + 1.09i)9-s + (0.705 + 0.0417i)10-s + (−0.251 − 0.167i)11-s + (−0.546 − 0.817i)12-s − 1.22·13-s + (−0.164 − 0.246i)14-s + (−0.269 − 1.94i)15-s + 0.250i·16-s + (−0.913 + 0.405i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.733 - 0.680i$
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: \((2,\ 170,\ (\ :1/2),\ 0.733 - 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45462 + 0.570867i\)
\(L(\frac12)\) \(\approx\) \(1.45462 + 0.570867i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.732 + 2.11i)T \)
17 \( 1 + (3.76 - 1.67i)T \)
good3 \( 1 + (-3.34 - 0.664i)T + (2.77 + 1.14i)T^{2} \)
7 \( 1 + (0.616 - 0.923i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (0.833 + 0.556i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + 4.43T + 13T^{2} \)
19 \( 1 + (-0.506 + 0.209i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.609 + 3.06i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (-0.588 + 2.96i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (1.23 - 0.826i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (1.12 - 5.66i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (1.75 + 8.82i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-0.0722 - 0.174i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 6.10iT - 47T^{2} \)
53 \( 1 + (-11.5 - 4.77i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-4.57 + 11.0i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-1.88 + 0.374i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + (-0.568 + 0.568i)T - 67iT^{2} \)
71 \( 1 + (2.52 + 3.78i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (1.01 + 1.51i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (2.90 - 4.35i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (4.60 - 11.1i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-4.95 - 4.95i)T + 89iT^{2} \)
97 \( 1 + (-3.75 - 5.62i)T + (-37.1 + 89.6i)T^{2} \)
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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

−13.19514219081857024730312753083, −12.30684808116849745965613583716, −10.33474181108283497081380463250, −9.430645248355544590812922052930, −8.716916946285455464865588769757, −8.061797658435093174444494571719, −7.04691213332864120419252107998, −5.02805309014832858646813723178, −3.98964272108206664459737031211, −2.31942541621268621327333577736, 2.22397937887716430784848156295, 3.11189349779223983426529951838, 4.24436199865432540658615704951, 7.05839982114922782321864617525, 7.48845873196839783733172918038, 8.645001090146861778362628155876, 9.659392063915623151325209704383, 10.31262315477955162610643026552, 11.73791265822223091909170153792, 12.87472754794440395486763301420

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