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Properties

Label 2-170-85.63-c1-0-7
Degree $2$
Conductor $170$
Sign $0.809 - 0.587i$
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 + (2.22 + 0.442i)3-s + (0.707 + 0.707i)4-s + (−2.23 − 0.151i)5-s + (1.88 + 1.26i)6-s + (−1.29 + 1.93i)7-s + (0.382 + 0.923i)8-s + (1.98 + 0.820i)9-s + (−2.00 − 0.993i)10-s + (1.86 − 2.78i)11-s + (1.26 + 1.88i)12-s − 4.73i·13-s + (−1.93 + 1.29i)14-s + (−4.89 − 1.32i)15-s + i·16-s + (−2.38 − 3.36i)17-s + ⋯
L(s)  = 1  + (0.653 + 0.270i)2-s + (1.28 + 0.255i)3-s + (0.353 + 0.353i)4-s + (−0.997 − 0.0677i)5-s + (0.769 + 0.514i)6-s + (−0.488 + 0.731i)7-s + (0.135 + 0.326i)8-s + (0.660 + 0.273i)9-s + (−0.633 − 0.314i)10-s + (0.561 − 0.840i)11-s + (0.363 + 0.544i)12-s − 1.31i·13-s + (−0.517 + 0.345i)14-s + (−1.26 − 0.341i)15-s + 0.250i·16-s + (−0.578 − 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.809 - 0.587i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.809 - 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84165 + 0.598368i\)
\(L(\frac12)\) \(\approx\) \(1.84165 + 0.598368i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.151i)T \)
17 \( 1 + (2.38 + 3.36i)T \)
good3 \( 1 + (-2.22 - 0.442i)T + (2.77 + 1.14i)T^{2} \)
7 \( 1 + (1.29 - 1.93i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (-1.86 + 2.78i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + 4.73iT - 13T^{2} \)
19 \( 1 + (0.786 - 0.325i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.820 - 4.12i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (3.83 + 0.762i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (-4.45 - 6.66i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (1.97 - 9.93i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (0.472 - 0.0940i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (1.69 - 0.702i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 6.52T + 47T^{2} \)
53 \( 1 + (1.33 - 3.22i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-5.52 + 13.3i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.17 + 5.90i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-0.912 - 0.912i)T + 67iT^{2} \)
71 \( 1 + (-10.9 + 7.33i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-6.77 - 10.1i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (6.77 + 4.52i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (-8.98 - 3.72i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-2.75 - 2.75i)T + 89iT^{2} \)
97 \( 1 + (-5.10 - 7.64i)T + (-37.1 + 89.6i)T^{2} \)
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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

−13.06949243232466328623981461422, −12.05144162684747411653747677642, −11.12554260362077296909596802253, −9.569909543159726963229556606200, −8.545152286351369513032897564569, −7.947502380965745963085479806865, −6.57980643998744719428883760454, −5.09550115874366027846193529090, −3.53464645156611179257062771608, −2.97084625111728967830771443197, 2.17161809543178633388177045178, 3.76015925729211120159307257457, 4.30068144099397489002780006071, 6.63876437114795644422938866588, 7.35439135503079463903444302565, 8.569783594700855363556149286097, 9.553983861690429916709552052481, 10.83826517501801376305601339019, 11.87723548467617622600407286098, 12.85857313168255122028732215247

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