↑

Properties

Label 2-85-85.4-c1-0-1
Degree $2$
Conductor $85$
Sign $0.994 + 0.105i$
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  − 1.80·2-s + (−0.397 − 0.397i)3-s + 1.25·4-s + (1.45 + 1.70i)5-s + (0.716 + 0.716i)6-s + (2.20 − 2.20i)7-s + 1.34·8-s − 2.68i·9-s + (−2.61 − 3.06i)10-s + (3.96 + 3.96i)11-s + (−0.497 − 0.497i)12-s + 1.24i·13-s + (−3.96 + 3.96i)14-s + (0.100 − 1.25i)15-s − 4.93·16-s + (−4.10 + 0.397i)17-s + ⋯
L(s)  = 1  − 1.27·2-s + (−0.229 − 0.229i)3-s + 0.626·4-s + (0.648 + 0.761i)5-s + (0.292 + 0.292i)6-s + (0.831 − 0.831i)7-s + 0.476·8-s − 0.894i·9-s + (−0.826 − 0.970i)10-s + (1.19 + 1.19i)11-s + (−0.143 − 0.143i)12-s + 0.346i·13-s + (−1.06 + 1.06i)14-s + (0.0258 − 0.323i)15-s − 1.23·16-s + (−0.995 + 0.0963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.994 + 0.105i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.994 + 0.105i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.585836 - 0.0310148i\)
\(L(\frac12)\) \(\approx\) \(0.585836 - 0.0310148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.45 - 1.70i)T \)
17 \( 1 + (4.10 - 0.397i)T \)
good2 \( 1 + 1.80T + 2T^{2} \)
3 \( 1 + (0.397 + 0.397i)T + 3iT^{2} \)
7 \( 1 + (-2.20 + 2.20i)T - 7iT^{2} \)
11 \( 1 + (-3.96 - 3.96i)T + 11iT^{2} \)
13 \( 1 - 1.24iT - 13T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-1.64 + 1.64i)T - 23iT^{2} \)
29 \( 1 + (4.68 - 4.68i)T - 29iT^{2} \)
31 \( 1 + (-3.22 + 3.22i)T - 31iT^{2} \)
37 \( 1 + (-1.34 - 1.34i)T + 37iT^{2} \)
41 \( 1 + (4.18 + 4.18i)T + 41iT^{2} \)
43 \( 1 + 2.04T + 43T^{2} \)
47 \( 1 + 4.85iT - 47T^{2} \)
53 \( 1 + 9.11T + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + (4 + 4i)T + 61iT^{2} \)
67 \( 1 - 8.46iT - 67T^{2} \)
71 \( 1 + (6.22 - 6.22i)T - 71iT^{2} \)
73 \( 1 + (1.10 + 1.10i)T + 73iT^{2} \)
79 \( 1 + (3.47 + 3.47i)T + 79iT^{2} \)
83 \( 1 - 3.94T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + (-8.76 - 8.76i)T + 97iT^{2} \)
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

−14.36682244443520524949302638016, −13.25323837933146945025779139944, −11.62579520094937940820092866679, −10.80195803270837395173500220869, −9.687541757469583450092601696269, −8.914701528210751132200443062749, −7.19275976516110972649965242320, −6.72531975240334504529577529635, −4.39453828749488269956903235870, −1.64281165254768694692459360562, 1.70819237065330717205640294444, 4.74774919234563207659906134334, 6.03326669581886786199472374977, 8.036189942773988481005235707069, 8.715914377069821221372367095793, 9.604301242313330369304009142387, 10.88016327763479187476078994970, 11.69129947696684217172496940543, 13.29287983323595709210858958745, 14.19925732040510690782090054189

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