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Properties

Label 2-170-85.73-c1-0-3
Degree $2$
Conductor $170$
Sign $-0.150 - 0.988i$
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 + (−0.664 + 3.34i)3-s + (0.707 + 0.707i)4-s + (1.48 − 1.67i)5-s + (−1.89 + 2.83i)6-s + (0.923 + 0.616i)7-s + (0.382 + 0.923i)8-s + (−7.94 − 3.29i)9-s + (2.01 − 0.976i)10-s + (−0.833 − 0.556i)11-s + (−2.83 + 1.89i)12-s − 4.43i·13-s + (0.616 + 0.923i)14-s + (4.59 + 6.07i)15-s + i·16-s + (1.67 + 3.76i)17-s + ⋯
L(s)  = 1  + (0.653 + 0.270i)2-s + (−0.383 + 1.92i)3-s + (0.353 + 0.353i)4-s + (0.664 − 0.747i)5-s + (−0.772 + 1.15i)6-s + (0.348 + 0.233i)7-s + (0.135 + 0.326i)8-s + (−2.64 − 1.09i)9-s + (0.636 − 0.308i)10-s + (−0.251 − 0.167i)11-s + (−0.817 + 0.546i)12-s − 1.22i·13-s + (0.164 + 0.246i)14-s + (1.18 + 1.56i)15-s + 0.250i·16-s + (0.405 + 0.913i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.150 - 0.988i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.150 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00785 + 1.17345i\)
\(L(\frac12)\) \(\approx\) \(1.00785 + 1.17345i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 1.67i)T \)
17 \( 1 + (-1.67 - 3.76i)T \)
good3 \( 1 + (0.664 - 3.34i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (-0.923 - 0.616i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (0.833 + 0.556i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + 4.43iT - 13T^{2} \)
19 \( 1 + (0.506 - 0.209i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-3.06 + 0.609i)T + (21.2 - 8.80i)T^{2} \)
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 \( 1 + (-5.66 - 1.12i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (1.75 + 8.82i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (0.174 - 0.0722i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 6.10T + 47T^{2} \)
53 \( 1 + (4.77 - 11.5i)T + (-37.4 - 37.4i)T^{2} \)
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 \( 1 + (-1.51 + 1.01i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (-2.90 + 4.35i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (11.1 + 4.60i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (4.95 + 4.95i)T + 89iT^{2} \)
97 \( 1 + (-5.62 + 3.75i)T + (37.1 - 89.6i)T^{2} \)
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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

−13.03955924666818027816514401706, −12.05571897546733072477741086235, −10.83592959738349483150434534436, −10.23739522632433131752097813814, −9.080070476874312901337779622385, −8.227330818991569467005717675508, −5.93391613127627765355764882725, −5.37208353351451741255655964885, −4.46897024390949874125654298261, −3.14624009989765734736490028863, 1.64145573430644786078128278065, 2.77529367333193796421277923005, 5.12524728993942100863741775879, 6.32131023828772460127203162741, 6.94662578879041404616873663834, 7.907072505836649108360657351120, 9.584473767828007215054234729072, 11.25310796616611996938244569894, 11.43580294739786387380877901976, 12.68986464757479674002348943946

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