↑

Properties

Label 2-71-71.5-c1-0-0
Degree $2$
Conductor $71$
Sign $-0.155 - 0.987i$
Analytic cond. $0.566937$
Root an. cond. $0.752952$
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  + (0.758 + 2.33i)2-s + (−0.252 − 0.777i)3-s + (−3.25 + 2.36i)4-s + (−0.0372 − 0.0270i)5-s + (1.62 − 1.17i)6-s + (0.331 + 1.02i)7-s + (−4.02 − 2.92i)8-s + (1.88 − 1.37i)9-s + (0.0349 − 0.107i)10-s + (−1.43 − 4.42i)11-s + (2.66 + 1.93i)12-s + (1.12 − 3.47i)13-s + (−2.13 + 1.55i)14-s + (−0.0116 + 0.0358i)15-s + (1.28 − 3.95i)16-s + (−2.33 + 7.17i)17-s + ⋯
L(s)  = 1  + (0.536 + 1.65i)2-s + (−0.145 − 0.448i)3-s + (−1.62 + 1.18i)4-s + (−0.0166 − 0.0121i)5-s + (0.662 − 0.481i)6-s + (0.125 + 0.386i)7-s + (−1.42 − 1.03i)8-s + (0.628 − 0.456i)9-s + (0.0110 − 0.0340i)10-s + (−0.433 − 1.33i)11-s + (0.768 + 0.558i)12-s + (0.313 − 0.963i)13-s + (−0.570 + 0.414i)14-s + (−0.00300 + 0.00925i)15-s + (0.321 − 0.989i)16-s + (−0.565 + 1.74i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(71\)
Sign: $-0.155 - 0.987i$
Analytic conductor: \(0.566937\)
Root analytic conductor: \(0.752952\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 71,\ (\ :1/2),\ -0.155 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.691062 + 0.808476i\)
\(L(\frac12)\) \(\approx\) \(0.691062 + 0.808476i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 + (0.618 + 8.40i)T \)
good2 \( 1 + (-0.758 - 2.33i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.252 + 0.777i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.0372 + 0.0270i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.331 - 1.02i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (1.43 + 4.42i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.12 + 3.47i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.33 - 7.17i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.693 - 2.13i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.80T + 23T^{2} \)
29 \( 1 + (-2.01 - 1.46i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.02 - 3.14i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 - 6.40T + 37T^{2} \)
41 \( 1 + 3.18T + 41T^{2} \)
43 \( 1 + (2.31 + 1.68i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (1.09 - 3.36i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-11.2 - 8.20i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (6.15 + 4.46i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.77 - 5.47i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.96 + 2.15i)T + (20.7 - 63.7i)T^{2} \)
73 \( 1 + (-4.05 - 12.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.376 + 0.273i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.07 - 5.13i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (6.76 + 4.91i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + 7.30T + 97T^{2} \)
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

−15.20651894939110723535376709747, −14.00113382683978908121707193620, −13.13402246927952069212576029520, −12.24018358670528113953824572761, −10.43416187319829569253180638232, −8.515546007422480641569773798017, −7.930359673483167001910888872407, −6.33662692812493795811082342028, −5.75904845988597855319117644091, −3.94190346216448895183352224993, 2.15743756779052185051154876231, 4.15823914812318433532250178085, 4.91975143698045273435028615818, 7.28671428327436522060792895510, 9.458373539765202368775906076525, 10.03858119896588640047379030349, 11.21098285956353658507720179878, 11.95390154634699634671302602378, 13.26419371730906857266147353175, 13.83476071001465958545377683619

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