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Properties

Label 2-170-17.16-c1-0-1
Degree $2$
Conductor $170$
Sign $0.242 - 0.970i$
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  + 2-s + 3i·3-s + 4-s + i·5-s + 3i·6-s − 4i·7-s + 8-s − 6·9-s + i·10-s + 2i·11-s + 3i·12-s + 13-s − 4i·14-s − 3·15-s + 16-s + (−4 − i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73i·3-s + 0.5·4-s + 0.447i·5-s + 1.22i·6-s − 1.51i·7-s + 0.353·8-s − 2·9-s + 0.316i·10-s + 0.603i·11-s + 0.866i·12-s + 0.277·13-s − 1.06i·14-s − 0.774·15-s + 0.250·16-s + (−0.970 − 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.242 - 0.970i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29796 + 1.01342i\)
\(L(\frac12)\) \(\approx\) \(1.29796 + 1.01342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - iT \)
17 \( 1 + (4 + i)T \)
good3 \( 1 - 3iT - 3T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + 7iT - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 8iT - 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 + 11T + 53T^{2} \)
59 \( 1 + 5T + 59T^{2} \)
61 \( 1 + iT - 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 - iT - 71T^{2} \)
73 \( 1 + 9iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 - iT - 97T^{2} \)
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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

−13.30155021929119161366332252046, −11.67626958654991442426548564908, −10.84344936356113208565349317874, −10.23045900052884807800347481520, −9.333687644170166982705742887466, −7.68616206475835792833706874124, −6.46767238726875828182751799868, −4.89538188592078517366076686232, −4.19873050916616556771392415588, −3.14050325071886133800571108651, 1.70330108837015106579750316817, 3.04581874417342468555685221023, 5.37230424282602979695769710811, 6.01869006498360600313377989302, 7.18802883405412538593069410516, 8.290761348738417328071218177964, 9.149973819366366417744698990508, 11.25111737501018010894859849339, 11.96265363250534614982759832729, 12.51617759768662949677062718120

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