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Properties

Label 2-170-85.27-c1-0-3
Degree $2$
Conductor $170$
Sign $0.957 - 0.287i$
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 + (2.85 − 0.567i)3-s + (0.707 − 0.707i)4-s + (−2.19 + 0.412i)5-s + (−2.41 + 1.61i)6-s + (2.40 + 3.60i)7-s + (−0.382 + 0.923i)8-s + (5.03 − 2.08i)9-s + (1.87 − 1.22i)10-s + (−0.113 − 0.170i)11-s + (1.61 − 2.41i)12-s − 3.32i·13-s + (−3.60 − 2.40i)14-s + (−6.03 + 2.42i)15-s − i·16-s + (−0.761 − 4.05i)17-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (1.64 − 0.327i)3-s + (0.353 − 0.353i)4-s + (−0.982 + 0.184i)5-s + (−0.986 + 0.659i)6-s + (0.910 + 1.36i)7-s + (−0.135 + 0.326i)8-s + (1.67 − 0.695i)9-s + (0.592 − 0.386i)10-s + (−0.0343 − 0.0513i)11-s + (0.466 − 0.697i)12-s − 0.923i·13-s + (−0.963 − 0.643i)14-s + (−1.55 + 0.625i)15-s − 0.250i·16-s + (−0.184 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.957 - 0.287i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.957 - 0.287i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28105 + 0.187978i\)
\(L(\frac12)\) \(\approx\) \(1.28105 + 0.187978i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 - 0.412i)T \)
17 \( 1 + (0.761 + 4.05i)T \)
good3 \( 1 + (-2.85 + 0.567i)T + (2.77 - 1.14i)T^{2} \)
7 \( 1 + (-2.40 - 3.60i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (0.113 + 0.170i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + 3.32iT - 13T^{2} \)
19 \( 1 + (1.76 + 0.729i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.153 - 0.769i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (8.95 - 1.78i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + (4.85 - 7.26i)T + (-11.8 - 28.6i)T^{2} \)
37 \( 1 + (0.374 + 1.88i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (3.42 + 0.682i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (6.29 + 2.60i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + (1.97 + 4.78i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.58 - 3.81i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-0.209 + 1.05i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-5.49 + 5.49i)T - 67iT^{2} \)
71 \( 1 + (1.33 + 0.892i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (0.452 - 0.677i)T + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (-10.3 + 6.88i)T + (30.2 - 72.9i)T^{2} \)
83 \( 1 + (0.650 - 0.269i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-3.70 + 3.70i)T - 89iT^{2} \)
97 \( 1 + (0.0834 - 0.124i)T + (-37.1 - 89.6i)T^{2} \)
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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

−12.80762697912406212710486739152, −11.84511947177461003621742658050, −10.76019471234675156445136864545, −9.199691643083633865712657421348, −8.663511451039728548703418722597, −7.88737364529413000728693801411, −7.14275638978868392658967728061, −5.22848097058486079944991422793, −3.34949285425138631853771647128, −2.13516337018683212748317101256, 1.85587811374525172252847517325, 3.77400353892020896752458312280, 4.24732813006133153122988233882, 7.15722820075592624037887959521, 7.85600943806744943705730413984, 8.528524284338645784090046650190, 9.520970849456650926878789158424, 10.65155404886340172483150251986, 11.42243849082895138527469607488, 12.87293010126922543413642062022

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