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Properties

Label 2-170-85.28-c1-0-4
Degree $2$
Conductor $170$
Sign $0.998 + 0.0459i$
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 + (0.530 − 0.793i)3-s + (0.707 − 0.707i)4-s + (1.12 + 1.93i)5-s + (−0.186 + 0.935i)6-s + (0.736 − 3.70i)7-s + (−0.382 + 0.923i)8-s + (0.799 + 1.93i)9-s + (−1.77 − 1.35i)10-s + (−2.74 − 0.546i)11-s + (−0.186 − 0.935i)12-s + 5.74·13-s + (0.736 + 3.70i)14-s + (2.12 + 0.134i)15-s − i·16-s + (3.33 − 2.41i)17-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (0.306 − 0.458i)3-s + (0.353 − 0.353i)4-s + (0.502 + 0.864i)5-s + (−0.0760 + 0.382i)6-s + (0.278 − 1.39i)7-s + (−0.135 + 0.326i)8-s + (0.266 + 0.643i)9-s + (−0.562 − 0.429i)10-s + (−0.828 − 0.164i)11-s + (−0.0537 − 0.270i)12-s + 1.59·13-s + (0.196 + 0.989i)14-s + (0.549 + 0.0346i)15-s − 0.250i·16-s + (0.809 − 0.586i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.998 + 0.0459i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.998 + 0.0459i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05736 - 0.0243283i\)
\(L(\frac12)\) \(\approx\) \(1.05736 - 0.0243283i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 - 1.93i)T \)
17 \( 1 + (-3.33 + 2.41i)T \)
good3 \( 1 + (-0.530 + 0.793i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (-0.736 + 3.70i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (2.74 + 0.546i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 - 5.74T + 13T^{2} \)
19 \( 1 + (-0.119 + 0.288i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (3.97 - 2.65i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-0.463 - 0.309i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (5.00 - 0.994i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (8.56 + 5.72i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (7.70 - 5.15i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (8.32 + 3.44i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 5.61iT - 47T^{2} \)
53 \( 1 + (2.52 + 6.08i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.14 + 1.30i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-5.67 - 8.48i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (-1.94 - 1.94i)T + 67iT^{2} \)
71 \( 1 + (-0.782 - 3.93i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (0.167 + 0.841i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (0.872 - 4.38i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (1.36 - 0.565i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-7.49 + 7.49i)T - 89iT^{2} \)
97 \( 1 + (3.04 + 15.3i)T + (-89.6 + 37.1i)T^{2} \)
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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

−13.19999675304607435433389698881, −11.35228651695531209549315207926, −10.56498144355848716499372079392, −10.01535487206668368854592817084, −8.405402409775016703644967932727, −7.54064894206833689992310230063, −6.83665611806579898168313582639, −5.44989244269331171472370548849, −3.45586909009926088648685004142, −1.64174203873616929073868792371, 1.79689201447626367069927299533, 3.53199209085652104087651577743, 5.24684941662289645761054905832, 6.28628774364791920230746803648, 8.341896765007664098083960321649, 8.611727019075350361494175131120, 9.668953021535564080477000731280, 10.50758533965703459953081420368, 11.92071569474059649314585225067, 12.52133543035360339895812744620

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