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Properties

Label 2-170-85.62-c1-0-6
Degree $2$
Conductor $170$
Sign $0.409 - 0.912i$
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 + (1.70 + 1.13i)3-s + (−0.707 + 0.707i)4-s + (2.03 − 0.937i)5-s + (−0.399 + 2.00i)6-s + (−1.27 − 0.252i)7-s + (−0.923 − 0.382i)8-s + (0.452 + 1.09i)9-s + (1.64 + 1.51i)10-s + (−2.68 − 0.534i)11-s + (−2.00 + 0.399i)12-s − 4.15i·13-s + (−0.252 − 1.27i)14-s + (4.51 + 0.713i)15-s − i·16-s + (−2.84 + 2.98i)17-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (0.981 + 0.656i)3-s + (−0.353 + 0.353i)4-s + (0.907 − 0.419i)5-s + (−0.162 + 0.818i)6-s + (−0.480 − 0.0956i)7-s + (−0.326 − 0.135i)8-s + (0.150 + 0.364i)9-s + (0.519 + 0.479i)10-s + (−0.810 − 0.161i)11-s + (−0.579 + 0.115i)12-s − 1.15i·13-s + (−0.0676 − 0.339i)14-s + (1.16 + 0.184i)15-s − 0.250i·16-s + (−0.690 + 0.723i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.409 - 0.912i$
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: \((2,\ 170,\ (\ :1/2),\ 0.409 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42688 + 0.923617i\)
\(L(\frac12)\) \(\approx\) \(1.42688 + 0.923617i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.03 + 0.937i)T \)
17 \( 1 + (2.84 - 2.98i)T \)
good3 \( 1 + (-1.70 - 1.13i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (1.27 + 0.252i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (2.68 + 0.534i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + 4.15iT - 13T^{2} \)
19 \( 1 + (2.69 - 6.51i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.92 + 2.87i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (-6.20 - 4.14i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (-0.796 + 0.158i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (1.73 - 2.59i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-7.15 + 4.78i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (0.813 - 1.96i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 0.904T + 47T^{2} \)
53 \( 1 + (-4.03 + 1.67i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-10.9 + 4.54i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.20 + 1.79i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (4.36 - 4.36i)T - 67iT^{2} \)
71 \( 1 + (-2.62 - 13.2i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (11.0 - 2.19i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (2.67 - 13.4i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (-2.47 - 5.96i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-1.83 + 1.83i)T - 89iT^{2} \)
97 \( 1 + (5.97 - 1.18i)T + (89.6 - 37.1i)T^{2} \)
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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

−13.05803212993086888284655294584, −12.55605366584631231007623816985, −10.39909906874125502182057987441, −9.954099377639108025941997459459, −8.615542556112416006603061998634, −8.195874464621116361221025460273, −6.44078925163149957721105483211, −5.44293111568472592203673226290, −4.04696568285287689319547660721, −2.71075134635928245920940744396, 2.15029180280356904088535277403, 2.84529881983808251672717196548, 4.71982181185200918358636220378, 6.31324467266590844328422494079, 7.31218637736294462830959427141, 8.813501967776582326040125146607, 9.475013318860675421314769884773, 10.59272277384061216571029403415, 11.65901882671159475381186242378, 13.01134637898344190337756739883

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