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Properties

Label 2-170-85.27-c1-0-1
Degree $2$
Conductor $170$
Sign $0.915 - 0.401i$
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 + (−3.22 + 0.641i)3-s + (0.707 − 0.707i)4-s + (2.10 + 0.761i)5-s + (−2.73 + 1.82i)6-s + (1.37 + 2.05i)7-s + (0.382 − 0.923i)8-s + (7.20 − 2.98i)9-s + (2.23 − 0.101i)10-s + (2.38 + 3.56i)11-s + (−1.82 + 2.73i)12-s + 1.04i·13-s + (2.05 + 1.37i)14-s + (−7.26 − 1.10i)15-s − i·16-s + (−3.07 − 2.75i)17-s + ⋯
L(s)  = 1  + (0.653 − 0.270i)2-s + (−1.86 + 0.370i)3-s + (0.353 − 0.353i)4-s + (0.940 + 0.340i)5-s + (−1.11 + 0.745i)6-s + (0.517 + 0.775i)7-s + (0.135 − 0.326i)8-s + (2.40 − 0.995i)9-s + (0.706 − 0.0320i)10-s + (0.719 + 1.07i)11-s + (−0.527 + 0.788i)12-s + 0.288i·13-s + (0.548 + 0.366i)14-s + (−1.87 − 0.285i)15-s − 0.250i·16-s + (−0.745 − 0.666i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.915 - 0.401i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.915 - 0.401i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14320 + 0.239874i\)
\(L(\frac12)\) \(\approx\) \(1.14320 + 0.239874i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.10 - 0.761i)T \)
17 \( 1 + (3.07 + 2.75i)T \)
good3 \( 1 + (3.22 - 0.641i)T + (2.77 - 1.14i)T^{2} \)
7 \( 1 + (-1.37 - 2.05i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (-2.38 - 3.56i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 - 1.04iT - 13T^{2} \)
19 \( 1 + (2.96 + 1.22i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.147 + 0.740i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (-0.174 + 0.0346i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + (-4.06 + 6.08i)T + (-11.8 - 28.6i)T^{2} \)
37 \( 1 + (-0.251 - 1.26i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (5.98 + 1.19i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (3.23 + 1.34i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 1.87T + 47T^{2} \)
53 \( 1 + (-1.33 - 3.21i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.48 + 8.40i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.70 - 8.57i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-11.5 + 11.5i)T - 67iT^{2} \)
71 \( 1 + (7.90 + 5.27i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (-5.61 + 8.39i)T + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (4.62 - 3.09i)T + (30.2 - 72.9i)T^{2} \)
83 \( 1 + (5.58 - 2.31i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (1.46 - 1.46i)T - 89iT^{2} \)
97 \( 1 + (0.783 - 1.17i)T + (-37.1 - 89.6i)T^{2} \)
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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

−12.52481225551036164386834440256, −11.78861326685402549016908721480, −11.11858230811863193980837594167, −10.13345881136138692377059127023, −9.306119546862556664980302172673, −6.87279320888559093398947523303, −6.27062062209602494040545549625, −5.17701714176645117422998202719, −4.44308506412541767431872514107, −1.94476810507122214413999358466, 1.37336869669521952382762384315, 4.26647937046916757228623592589, 5.29990596252998059027763725218, 6.20984246101819543598135578206, 6.84433037046586154808022898332, 8.405933590393400269738242980794, 10.22343554454621194157427468249, 10.92290465556846781083986445931, 11.74476009223070123409173343008, 12.77946089073462538118151448347

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