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Properties

Label 2-170-85.82-c1-0-2
Degree $2$
Conductor $170$
Sign $0.621 - 0.783i$
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 + (1.63 + 2.44i)3-s + (0.707 + 0.707i)4-s + (2.09 + 0.778i)5-s + (−0.574 − 2.88i)6-s + (−0.525 − 2.63i)7-s + (−0.382 − 0.923i)8-s + (−2.17 + 5.24i)9-s + (−1.63 − 1.52i)10-s + (0.617 − 0.122i)11-s + (−0.574 + 2.88i)12-s − 3.89·13-s + (−0.525 + 2.63i)14-s + (1.52 + 6.40i)15-s + i·16-s + (−0.231 − 4.11i)17-s + ⋯
L(s)  = 1  + (−0.653 − 0.270i)2-s + (0.944 + 1.41i)3-s + (0.353 + 0.353i)4-s + (0.937 + 0.348i)5-s + (−0.234 − 1.17i)6-s + (−0.198 − 0.997i)7-s + (−0.135 − 0.326i)8-s + (−0.723 + 1.74i)9-s + (−0.518 − 0.481i)10-s + (0.186 − 0.0370i)11-s + (−0.165 + 0.833i)12-s − 1.08·13-s + (−0.140 + 0.705i)14-s + (0.393 + 1.65i)15-s + 0.250i·16-s + (−0.0561 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.621 - 0.783i$
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: \((2,\ 170,\ (\ :1/2),\ 0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10230 + 0.532262i\)
\(L(\frac12)\) \(\approx\) \(1.10230 + 0.532262i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.09 - 0.778i)T \)
17 \( 1 + (0.231 + 4.11i)T \)
good3 \( 1 + (-1.63 - 2.44i)T + (-1.14 + 2.77i)T^{2} \)
7 \( 1 + (0.525 + 2.63i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.617 + 0.122i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + 3.89T + 13T^{2} \)
19 \( 1 + (-1.11 - 2.68i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.71 + 1.14i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (-7.22 + 4.82i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (8.76 + 1.74i)T + (28.6 + 11.8i)T^{2} \)
37 \( 1 + (2.69 - 1.80i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-3.52 - 2.35i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (8.90 - 3.68i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 1.02iT - 47T^{2} \)
53 \( 1 + (-5.45 + 13.1i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.84 + 1.17i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-7.66 + 11.4i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (-2.36 + 2.36i)T - 67iT^{2} \)
71 \( 1 + (2.62 - 13.1i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (1.65 - 8.34i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (0.558 + 2.80i)T + (-72.9 + 30.2i)T^{2} \)
83 \( 1 + (1.59 + 0.661i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-5.74 - 5.74i)T + 89iT^{2} \)
97 \( 1 + (-0.528 + 2.65i)T + (-89.6 - 37.1i)T^{2} \)
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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

−13.13747238377097996381954479888, −11.51813956626463086244270135572, −10.31204425169092704162791858338, −9.943738764992801211300873555425, −9.297678250899689251845573646075, −8.055497681688255061346227910562, −6.85903406152188451470262181950, −5.03161315304250299470401241783, −3.67880551813862079835367626800, −2.46965577099727423305049620770, 1.69798641318551952420143151547, 2.70071794126752649820138583222, 5.48314101189262885949047959191, 6.56044328090932582064076379220, 7.47678601235428642667525735819, 8.767432215702445517511681466634, 9.039021136246432411920985990479, 10.31496819791640292906889831292, 12.09972316134756534044245407662, 12.58175568819692295498331135551

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