Properties

Label 2-170-5.4-c1-0-5
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  + i·2-s − 3.14i·3-s − 4-s + (1.38 + 1.75i)5-s + 3.14·6-s − 3.50i·7-s i·8-s − 6.86·9-s + (−1.75 + 1.38i)10-s + 2·11-s + 3.14i·12-s − 3.14i·13-s + 3.50·14-s + (5.50 − 4.36i)15-s + 16-s + i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.81i·3-s − 0.5·4-s + (0.621 + 0.783i)5-s + 1.28·6-s − 1.32i·7-s − 0.353i·8-s − 2.28·9-s + (−0.554 + 0.439i)10-s + 0.603·11-s + 0.906i·12-s − 0.871i·13-s + 0.936·14-s + (1.42 − 1.12i)15-s + 0.250·16-s + 0.242i·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: $\chi_{170} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.621 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04755 - 0.506381i\)
\(L(\frac12)\) \(\approx\) \(1.04755 - 0.506381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 - 1.75i)T \)
17 \( 1 - iT \)
good3 \( 1 + 3.14iT - 3T^{2} \)
7 \( 1 + 3.50iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 3.14iT - 13T^{2} \)
19 \( 1 - 3.86T + 19T^{2} \)
23 \( 1 - 7.50iT - 23T^{2} \)
29 \( 1 - 0.646T + 29T^{2} \)
31 \( 1 + 3.91T + 31T^{2} \)
37 \( 1 - 8.95iT - 37T^{2} \)
41 \( 1 - 6.72T + 41T^{2} \)
43 \( 1 - 4.28iT - 43T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 - 8.69iT - 53T^{2} \)
59 \( 1 + 4.41T + 59T^{2} \)
61 \( 1 + 0.910T + 61T^{2} \)
67 \( 1 + 10.2iT - 67T^{2} \)
71 \( 1 - 3.19T + 71T^{2} \)
73 \( 1 + 3.14iT - 73T^{2} \)
79 \( 1 + 2.05T + 79T^{2} \)
83 \( 1 + 1.73iT - 83T^{2} \)
89 \( 1 - 4.41T + 89T^{2} \)
97 \( 1 - 2.41iT - 97T^{2} \)
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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

−13.07658404310482880861235532353, −11.81903851695595316446881693894, −10.71674071170474310931172018317, −9.459567589321122516220892460424, −7.888810606401687350307816774916, −7.33753172193670558649994721704, −6.55730885033871885714749563519, −5.60140877042469304893742287704, −3.30899342325184736431873130847, −1.30711590399809800833614645236, 2.49878211766599260146951220993, 4.07353802233461188795951944979, 5.03779651211742696822438279832, 5.92852871120185376302129876859, 8.633607304282267166189177547652, 9.185770991845467508306027377481, 9.680971894598569644055726042676, 10.86973855501394886258207268557, 11.78527674172910679088883580801, 12.59571838914073293247337322614

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