↑

Properties

Label 2-170-85.22-c1-0-0
Degree $2$
Conductor $170$
Sign $-0.935 + 0.354i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 + 0.923i)2-s + (−1.27 − 0.253i)3-s + (−0.707 − 0.707i)4-s + (−1.36 + 1.77i)5-s + (0.722 − 1.08i)6-s + (−0.128 + 0.192i)7-s + (0.923 − 0.382i)8-s + (−1.21 − 0.501i)9-s + (−1.11 − 1.93i)10-s + (−3.12 − 2.08i)11-s + (0.722 + 1.08i)12-s − 6.31·13-s + (−0.128 − 0.192i)14-s + (2.18 − 1.91i)15-s + i·16-s + (−2.15 + 3.51i)17-s + ⋯
L(s)  = 1  + (−0.270 + 0.653i)2-s + (−0.735 − 0.146i)3-s + (−0.353 − 0.353i)4-s + (−0.609 + 0.792i)5-s + (0.294 − 0.441i)6-s + (−0.0486 + 0.0728i)7-s + (0.326 − 0.135i)8-s + (−0.403 − 0.167i)9-s + (−0.352 − 0.612i)10-s + (−0.941 − 0.629i)11-s + (0.208 + 0.311i)12-s − 1.75·13-s + (−0.0344 − 0.0514i)14-s + (0.564 − 0.494i)15-s + 0.250i·16-s + (−0.522 + 0.852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.935 + 0.354i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.935 + 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0270941 - 0.147959i\)
\(L(\frac12)\) \(\approx\) \(0.0270941 - 0.147959i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 + (1.36 - 1.77i)T \)
17 \( 1 + (2.15 - 3.51i)T \)
good3 \( 1 + (1.27 + 0.253i)T + (2.77 + 1.14i)T^{2} \)
7 \( 1 + (0.128 - 0.192i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (3.12 + 2.08i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + 6.31T + 13T^{2} \)
19 \( 1 + (-5.28 + 2.18i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.67 - 8.43i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (-0.221 + 1.11i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (1.28 - 0.856i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (-0.0492 + 0.247i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-1.06 - 5.36i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (3.68 + 8.89i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 4.58iT - 47T^{2} \)
53 \( 1 + (5.77 + 2.39i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (0.547 - 1.32i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (10.1 - 2.02i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + (0.973 - 0.973i)T - 67iT^{2} \)
71 \( 1 + (1.06 + 1.59i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (1.07 + 1.60i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (-0.466 + 0.698i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (-5.53 + 13.3i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (9.96 + 9.96i)T + 89iT^{2} \)
97 \( 1 + (-2.79 - 4.18i)T + (-37.1 + 89.6i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.41156010374559298207249714792, −12.13348642452547535749554405556, −11.31822771318031225894253184562, −10.43451357694445455169809746179, −9.265312783568351146223411943816, −7.84113262829201099901949809303, −7.17037616051431025835998291573, −5.95402320943128959401553594023, −4.98401394895833610788359854230, −3.05876507751151419401197867131, 0.15728281895869833293073069333, 2.66288771691891477173600793561, 4.67066966545173751640802662183, 5.18721042707378425162113211566, 7.18734347589421207467362734084, 8.146380735905079619423548701818, 9.397864399611410904712821926263, 10.29844561895748051520254859437, 11.34087443350351077871689890683, 12.19069371399064500452754588363

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