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Properties

Label 2-170-85.3-c1-0-6
Degree $2$
Conductor $170$
Sign $0.0945 + 0.995i$
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 + (−0.419 − 0.628i)3-s + (−0.707 − 0.707i)4-s + (2.19 + 0.423i)5-s + (−0.741 + 0.147i)6-s + (−0.252 − 1.27i)7-s + (−0.923 + 0.382i)8-s + (0.929 − 2.24i)9-s + (1.23 − 1.86i)10-s + (−0.546 − 2.74i)11-s + (−0.147 + 0.741i)12-s + 4.10i·13-s + (−1.27 − 0.252i)14-s + (−0.655 − 1.55i)15-s + i·16-s + (−3.27 + 2.50i)17-s + ⋯
L(s)  = 1  + (0.270 − 0.653i)2-s + (−0.242 − 0.362i)3-s + (−0.353 − 0.353i)4-s + (0.981 + 0.189i)5-s + (−0.302 + 0.0601i)6-s + (−0.0955 − 0.480i)7-s + (−0.326 + 0.135i)8-s + (0.309 − 0.747i)9-s + (0.389 − 0.590i)10-s + (−0.164 − 0.827i)11-s + (−0.0425 + 0.213i)12-s + 1.13i·13-s + (−0.339 − 0.0675i)14-s + (−0.169 − 0.402i)15-s + 0.250i·16-s + (−0.794 + 0.607i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0945 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0945 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.0945 + 0.995i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.0945 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.976991 - 0.888567i\)
\(L(\frac12)\) \(\approx\) \(0.976991 - 0.888567i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 - 0.423i)T \)
17 \( 1 + (3.27 - 2.50i)T \)
good3 \( 1 + (0.419 + 0.628i)T + (-1.14 + 2.77i)T^{2} \)
7 \( 1 + (0.252 + 1.27i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (0.546 + 2.74i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 - 4.10iT - 13T^{2} \)
19 \( 1 + (1.17 + 2.83i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-6.28 - 4.19i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (-3.51 - 5.26i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (0.355 - 1.78i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-0.284 + 0.190i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (1.73 - 2.59i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (-3.04 - 7.35i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 7.90T + 47T^{2} \)
53 \( 1 + (10.3 + 4.29i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (9.44 + 3.91i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-5.15 - 3.44i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (0.944 + 0.944i)T + 67iT^{2} \)
71 \( 1 + (7.94 + 1.58i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (0.349 - 1.75i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-16.2 + 3.23i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (-1.37 + 3.31i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (7.01 + 7.01i)T + 89iT^{2} \)
97 \( 1 + (-1.63 + 8.19i)T + (-89.6 - 37.1i)T^{2} \)
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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

−12.74672362136269610635739356478, −11.41367919667269743648474165140, −10.78492055137418827797090062401, −9.569615097214727647035000937837, −8.839948254593424306569491494037, −6.90848701544903956149951673485, −6.22066396957525092478391881638, −4.76316797770487805003926547339, −3.22447778383863703174652977608, −1.47976387506554244801196647631, 2.49431628069024659455124833320, 4.62408496261675693126268806785, 5.38238589521256929916072810515, 6.49631374201015677432233535980, 7.77050403510670383189620438371, 8.961506508203808010820491667163, 9.973330736642176191189722956361, 10.79534124181877497678819421465, 12.38474628450518350222052580425, 13.07151173220185002464942156501

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