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Properties

Label 2-170-85.62-c1-0-2
Degree $2$
Conductor $170$
Sign $0.999 - 0.0214i$
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 + (1.25 + 0.839i)3-s + (−0.707 + 0.707i)4-s + (−1.93 + 1.11i)5-s + (0.294 − 1.48i)6-s + (4.13 + 0.823i)7-s + (0.923 + 0.382i)8-s + (−0.273 − 0.660i)9-s + (1.77 + 1.36i)10-s + (4.92 + 0.980i)11-s + (−1.48 + 0.294i)12-s + 1.01i·13-s + (−0.823 − 4.13i)14-s + (−3.37 − 0.221i)15-s − i·16-s + (−3.78 − 1.64i)17-s + ⋯
L(s)  = 1  + (−0.270 − 0.653i)2-s + (0.725 + 0.484i)3-s + (−0.353 + 0.353i)4-s + (−0.866 + 0.499i)5-s + (0.120 − 0.605i)6-s + (1.56 + 0.311i)7-s + (0.326 + 0.135i)8-s + (−0.0912 − 0.220i)9-s + (0.560 + 0.430i)10-s + (1.48 + 0.295i)11-s + (−0.427 + 0.0851i)12-s + 0.282i·13-s + (−0.219 − 1.10i)14-s + (−0.870 − 0.0571i)15-s − 0.250i·16-s + (−0.917 − 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.999 - 0.0214i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.999 - 0.0214i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21991 + 0.0130675i\)
\(L(\frac12)\) \(\approx\) \(1.21991 + 0.0130675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.93 - 1.11i)T \)
17 \( 1 + (3.78 + 1.64i)T \)
good3 \( 1 + (-1.25 - 0.839i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (-4.13 - 0.823i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (-4.92 - 0.980i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 - 1.01iT - 13T^{2} \)
19 \( 1 + (2.35 - 5.67i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.21 + 6.31i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (0.994 + 0.664i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (1.71 - 0.341i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (-0.639 + 0.956i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (3.25 - 2.17i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-2.10 + 5.07i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + (4.29 - 1.77i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-10.1 + 4.18i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.916 + 1.37i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (-4.09 + 4.09i)T - 67iT^{2} \)
71 \( 1 + (-1.92 - 9.69i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-10.3 + 2.05i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (0.444 - 2.23i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (1.59 + 3.85i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-1.31 + 1.31i)T - 89iT^{2} \)
97 \( 1 + (12.3 - 2.46i)T + (89.6 - 37.1i)T^{2} \)
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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

−12.31495610138792096180474155032, −11.69284065024600645928682456444, −10.95043837682197000138460688488, −9.725313150318217917025509757658, −8.600308471109717086702692328763, −8.147628533279489023970600865580, −6.62607984513576801954874572432, −4.46758026862006616808671553260, −3.79149588435751002781643613913, −2.07247990432937950606566200724, 1.58636239270564518282773100888, 3.98928228322470585381014698896, 5.06200410992192609540823154873, 6.81382511724816774167528561008, 7.83513105385499368448542264662, 8.416192136263367784128745925841, 9.182388680097805993061649765438, 11.07155418926101913473902419667, 11.56547887096010030516029296688, 13.04997442160508850179808042904

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