↑

Properties

Label 2-85-17.9-c1-0-2
Degree $2$
Conductor $85$
Sign $0.789 + 0.614i$
Analytic cond. $0.678728$
Root an. cond. $0.823849$
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  + (−1.93 + 1.93i)2-s + (−1.42 − 0.591i)3-s − 5.45i·4-s + (0.382 − 0.923i)5-s + (3.90 − 1.61i)6-s + (−0.483 − 1.16i)7-s + (6.67 + 6.67i)8-s + (−0.429 − 0.429i)9-s + (1.04 + 2.52i)10-s + (0.386 − 0.159i)11-s + (−3.22 + 7.79i)12-s − 5.66i·13-s + (3.18 + 1.32i)14-s + (−1.09 + 1.09i)15-s − 14.8·16-s + (1.27 − 3.92i)17-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)2-s + (−0.825 − 0.341i)3-s − 2.72i·4-s + (0.171 − 0.413i)5-s + (1.59 − 0.659i)6-s + (−0.182 − 0.441i)7-s + (2.35 + 2.35i)8-s + (−0.143 − 0.143i)9-s + (0.330 + 0.797i)10-s + (0.116 − 0.0482i)11-s + (−0.932 + 2.25i)12-s − 1.57i·13-s + (0.852 + 0.353i)14-s + (−0.282 + 0.282i)15-s − 3.71·16-s + (0.309 − 0.950i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.789 + 0.614i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.789 + 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305878 - 0.104999i\)
\(L(\frac12)\) \(\approx\) \(0.305878 - 0.104999i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (-1.27 + 3.92i)T \)
good2 \( 1 + (1.93 - 1.93i)T - 2iT^{2} \)
3 \( 1 + (1.42 + 0.591i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (0.483 + 1.16i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.386 + 0.159i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 5.66iT - 13T^{2} \)
19 \( 1 + (0.0948 - 0.0948i)T - 19iT^{2} \)
23 \( 1 + (6.30 - 2.60i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.126 - 0.304i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (0.559 + 0.231i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-8.16 - 3.38i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.625 - 1.50i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.41 + 1.41i)T + 43iT^{2} \)
47 \( 1 + 5.06iT - 47T^{2} \)
53 \( 1 + (1.09 - 1.09i)T - 53iT^{2} \)
59 \( 1 + (-0.997 - 0.997i)T + 59iT^{2} \)
61 \( 1 + (-2.98 - 7.21i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + (4.45 + 1.84i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (1.10 - 2.67i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-4.00 + 1.65i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-3.10 + 3.10i)T - 83iT^{2} \)
89 \( 1 + 4.98iT - 89T^{2} \)
97 \( 1 + (-0.243 + 0.587i)T + (-68.5 - 68.5i)T^{2} \)
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

−14.51266238428336667858855354922, −13.30032005719569766266001815068, −11.71157898823074063664904482539, −10.44048439188333063627250073768, −9.589131899389589947362526283906, −8.287211704614109394575445016164, −7.30918999068698331973787007628, −6.09605228828348968395098163212, −5.30792889446107921951414610202, −0.69569041440203107906335757608, 2.21195337879718829362405162580, 4.09793045906188679932659025571, 6.36791281353459152180680864396, 8.001759764439202344084297525463, 9.207679015789890461103200196241, 10.12672219570034267159152919679, 11.05458141973646286076921162224, 11.74897272786230571670296904970, 12.63488691906418355366101202956, 14.14936269376982037164473616966

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