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Properties

Label 2-170-85.82-c1-0-1
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: $\chi_{170} (167, \cdot )$
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

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

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