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Properties

Label 2-170-17.13-c1-0-0
Degree $2$
Conductor $170$
Sign $-0.869 - 0.493i$
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 + (−2.31 − 2.31i)3-s − 4-s + (0.707 + 0.707i)5-s + (2.31 − 2.31i)6-s + (−3.26 + 3.26i)7-s − i·8-s + 7.68i·9-s + (−0.707 + 0.707i)10-s + (−1.82 + 1.82i)11-s + (2.31 + 2.31i)12-s − 0.145·13-s + (−3.26 − 3.26i)14-s − 3.26i·15-s + 16-s + (−3.31 − 2.45i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−1.33 − 1.33i)3-s − 0.5·4-s + (0.316 + 0.316i)5-s + (0.943 − 0.943i)6-s + (−1.23 + 1.23i)7-s − 0.353i·8-s + 2.56i·9-s + (−0.223 + 0.223i)10-s + (−0.551 + 0.551i)11-s + (0.667 + 0.667i)12-s − 0.0404·13-s + (−0.873 − 0.873i)14-s − 0.843i·15-s + 0.250·16-s + (−0.803 − 0.595i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.869 - 0.493i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.869 - 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0673412 + 0.255256i\)
\(L(\frac12)\) \(\approx\) \(0.0673412 + 0.255256i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (3.31 + 2.45i)T \)
good3 \( 1 + (2.31 + 2.31i)T + 3iT^{2} \)
7 \( 1 + (3.26 - 3.26i)T - 7iT^{2} \)
11 \( 1 + (1.82 - 1.82i)T - 11iT^{2} \)
13 \( 1 + 0.145T + 13T^{2} \)
19 \( 1 + 2.06iT - 19T^{2} \)
23 \( 1 + (3.41 - 3.41i)T - 23iT^{2} \)
29 \( 1 + (-2.10 - 2.10i)T + 29iT^{2} \)
31 \( 1 + (2.78 + 2.78i)T + 31iT^{2} \)
37 \( 1 + (2.44 + 2.44i)T + 37iT^{2} \)
41 \( 1 + (5.53 - 5.53i)T - 41iT^{2} \)
43 \( 1 - 0.622iT - 43T^{2} \)
47 \( 1 - 8.47T + 47T^{2} \)
53 \( 1 - 6.68iT - 53T^{2} \)
59 \( 1 - 5.71iT - 59T^{2} \)
61 \( 1 + (-2.63 + 2.63i)T - 61iT^{2} \)
67 \( 1 - 1.58T + 67T^{2} \)
71 \( 1 + (-1.83 - 1.83i)T + 71iT^{2} \)
73 \( 1 + (5.82 + 5.82i)T + 73iT^{2} \)
79 \( 1 + (-3.29 + 3.29i)T - 79iT^{2} \)
83 \( 1 - 8.91iT - 83T^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 + (-12.5 - 12.5i)T + 97iT^{2} \)
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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

−13.05269884870179356363748153236, −12.38942882626158973099923644901, −11.51385533592454277287580482282, −10.21914060837716173961962905069, −9.030539118189100406369247266524, −7.54791452025430575082422008114, −6.70550863322512808911574917760, −5.97755222760080624921898391196, −5.14358540545905003732702264148, −2.39996727209356729416308901166, 0.27360264928288944835914032629, 3.54032759469673130085499584802, 4.39720823591526301814656202837, 5.66899008268740755221314172105, 6.66061955354920187109510148369, 8.758275819675845773273496043610, 9.986476698140121761771435950842, 10.29294011252670909482008716678, 11.05427545832356810061472749089, 12.23159075977515129925938664309

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