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Properties

Label 2-170-85.57-c1-0-6
Degree $2$
Conductor $170$
Sign $0.978 - 0.207i$
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 + (1.42 − 2.12i)3-s + (−0.707 + 0.707i)4-s + (0.986 + 2.00i)5-s + (2.51 + 0.499i)6-s + (−0.00818 + 0.0411i)7-s + (−0.923 − 0.382i)8-s + (−1.36 − 3.29i)9-s + (−1.47 + 1.67i)10-s + (0.476 − 2.39i)11-s + (0.499 + 2.51i)12-s + 2.17i·13-s + (−0.0411 + 0.00818i)14-s + (5.67 + 0.754i)15-s − i·16-s + (−3.36 − 2.37i)17-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (0.821 − 1.22i)3-s + (−0.353 + 0.353i)4-s + (0.441 + 0.897i)5-s + (1.02 + 0.204i)6-s + (−0.00309 + 0.0155i)7-s + (−0.326 − 0.135i)8-s + (−0.454 − 1.09i)9-s + (−0.466 + 0.531i)10-s + (0.143 − 0.721i)11-s + (0.144 + 0.725i)12-s + 0.601i·13-s + (−0.0109 + 0.00218i)14-s + (1.46 + 0.194i)15-s − 0.250i·16-s + (−0.816 − 0.576i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.978 - 0.207i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.978 - 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62205 + 0.170301i\)
\(L(\frac12)\) \(\approx\) \(1.62205 + 0.170301i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.986 - 2.00i)T \)
17 \( 1 + (3.36 + 2.37i)T \)
good3 \( 1 + (-1.42 + 2.12i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (0.00818 - 0.0411i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (-0.476 + 2.39i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 - 2.17iT - 13T^{2} \)
19 \( 1 + (-1.43 + 3.45i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (5.93 - 3.96i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (4.73 - 7.08i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-0.226 - 1.13i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (6.10 + 4.07i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-0.400 - 0.599i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-3.46 + 8.37i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 4.13T + 47T^{2} \)
53 \( 1 + (-4.87 + 2.01i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-8.77 + 3.63i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-5.87 + 3.92i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (11.0 - 11.0i)T - 67iT^{2} \)
71 \( 1 + (-5.22 + 1.04i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (2.22 + 11.1i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (-7.04 - 1.40i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (-6.46 - 15.6i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (11.5 - 11.5i)T - 89iT^{2} \)
97 \( 1 + (0.800 + 4.02i)T + (-89.6 + 37.1i)T^{2} \)
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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

−13.28752010355994193474817857206, −12.05868363866228471220137720155, −10.98350532553067123277064223498, −9.382496758001900141128470177904, −8.534218583960429856258344896155, −7.23801578205693990931547255815, −6.86044595337699741786553881313, −5.60676883660926029547264467932, −3.53567889801617007923693117776, −2.18160664151867654750901983738, 2.19250895029967837208629133621, 3.86600207490177990870319561125, 4.62004189955558387937041134913, 5.90240686607232915482828687391, 8.087408866137440601775962576207, 8.948606158233275247766890891323, 9.910636864104921178970883635667, 10.32131546115021931504759642108, 11.81395768627252252078540137561, 12.80070867784557779101759332156

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