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Properties

Label 2-170-85.57-c1-0-5
Degree $2$
Conductor $170$
Sign $-0.0801 + 0.996i$
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 + (−0.800 + 1.19i)3-s + (−0.707 + 0.707i)4-s + (−0.598 − 2.15i)5-s + (1.41 + 0.281i)6-s + (0.660 − 3.32i)7-s + (0.923 + 0.382i)8-s + (0.352 + 0.851i)9-s + (−1.76 + 1.37i)10-s + (0.778 − 3.91i)11-s + (−0.281 − 1.41i)12-s − 4.10i·13-s + (−3.32 + 0.660i)14-s + (3.06 + 1.00i)15-s − i·16-s + (2.71 − 3.10i)17-s + ⋯
L(s)  = 1  + (−0.270 − 0.653i)2-s + (−0.462 + 0.692i)3-s + (−0.353 + 0.353i)4-s + (−0.267 − 0.963i)5-s + (0.577 + 0.114i)6-s + (0.249 − 1.25i)7-s + (0.326 + 0.135i)8-s + (0.117 + 0.283i)9-s + (−0.556 + 0.435i)10-s + (0.234 − 1.18i)11-s + (−0.0811 − 0.408i)12-s − 1.13i·13-s + (−0.888 + 0.176i)14-s + (0.790 + 0.260i)15-s − 0.250i·16-s + (0.658 − 0.752i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0801 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0801 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.0801 + 0.996i$
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: \((2,\ 170,\ (\ :1/2),\ -0.0801 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.533888 - 0.578519i\)
\(L(\frac12)\) \(\approx\) \(0.533888 - 0.578519i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.598 + 2.15i)T \)
17 \( 1 + (-2.71 + 3.10i)T \)
good3 \( 1 + (0.800 - 1.19i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (-0.660 + 3.32i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (-0.778 + 3.91i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + 4.10iT - 13T^{2} \)
19 \( 1 + (2.45 - 5.92i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (2.73 - 1.82i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-4.93 + 7.38i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-2.05 - 10.3i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-5.26 - 3.51i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-0.637 - 0.953i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-1.76 + 4.26i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 0.236T + 47T^{2} \)
53 \( 1 + (-0.849 + 0.352i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.06 + 1.27i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.930 + 0.621i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (6.65 - 6.65i)T - 67iT^{2} \)
71 \( 1 + (-6.11 + 1.21i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (0.411 + 2.06i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (-0.0457 - 0.00909i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (-0.376 - 0.909i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (6.70 - 6.70i)T - 89iT^{2} \)
97 \( 1 + (-1.36 - 6.84i)T + (-89.6 + 37.1i)T^{2} \)
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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

−12.29570855229482038745985018704, −11.39259232475016301915726384091, −10.40848144983424305494106902533, −9.929388268788972471096988390518, −8.384889783526243405443647598263, −7.75639358456395590820549754647, −5.70587509037498640675097510750, −4.57210269972873645878126612894, −3.58104195804689148633787143295, −0.920825973204736088880197454549, 2.16998163112923751626031035144, 4.38092043780840030906055403590, 6.01972923021615607817139299389, 6.69210188849328837613034690396, 7.58621204454291269429607990096, 8.892779801325697537817453576513, 9.860344850814158939393625644619, 11.25169539670199191030187447279, 12.04118596523997984193391521688, 12.85129533774406697859242915323

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