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Properties

Label 2-170-85.84-c1-0-3
Degree $2$
Conductor $170$
Sign $0.420 - 0.907i$
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  + i·2-s + 2.82·3-s − 4-s + (0.707 + 2.12i)5-s + 2.82i·6-s − 4.24·7-s − i·8-s + 5.00·9-s + (−2.12 + 0.707i)10-s − 2.82·12-s − 6i·13-s − 4.24i·14-s + (2.00 + 6i)15-s + 16-s + (2.82 − 3i)17-s + 5.00i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.63·3-s − 0.5·4-s + (0.316 + 0.948i)5-s + 1.15i·6-s − 1.60·7-s − 0.353i·8-s + 1.66·9-s + (−0.670 + 0.223i)10-s − 0.816·12-s − 1.66i·13-s − 1.13i·14-s + (0.516 + 1.54i)15-s + 0.250·16-s + (0.685 − 0.727i)17-s + 1.17i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.420 - 0.907i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.420 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36552 + 0.871965i\)
\(L(\frac12)\) \(\approx\) \(1.36552 + 0.871965i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-0.707 - 2.12i)T \)
17 \( 1 + (-2.82 + 3i)T \)
good3 \( 1 - 2.82T + 3T^{2} \)
7 \( 1 + 4.24T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 - 4.24iT - 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 4.24iT - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 4.24iT - 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 + 4.24iT - 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−13.30956120658930436056588348478, −12.49374556968304931490314759437, −10.33506300331523678719266673261, −9.804568842318255675860229131819, −8.860554645954631556466625401263, −7.65822435266637210201028614554, −6.95699127394199851292202224631, −5.66876859062903956842955748051, −3.40772590844707343908426910388, −2.97024208037910435614385931099, 1.91505430413316352158022021370, 3.31581171376736480719605291617, 4.28697831313215297950563479453, 6.23266659380700618461353091116, 7.77265927397674281762515943603, 8.956520327368728494365539279328, 9.418839338644338202652010668407, 10.08505581735487839464065369489, 11.95466595037747718627858757914, 12.76340090201164043951785380031

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