↑

Properties

Label 2-4020-67.37-c1-0-27
Degree $2$
Conductor $4020$
Sign $0.686 + 0.727i$
Analytic cond. $32.0998$
Root an. cond. $5.66567$
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  + 3-s − 5-s + (−1.68 + 2.92i)7-s + 9-s + (−0.686 + 1.18i)11-s + (0.5 + 0.866i)13-s − 15-s + (−3.68 − 6.38i)17-s + (−0.313 − 0.543i)19-s + (−1.68 + 2.92i)21-s + (−3.68 − 6.38i)23-s + 25-s + 27-s + (0.686 − 1.18i)29-s + (1.87 − 3.24i)31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + (−0.637 + 1.10i)7-s + 0.333·9-s + (−0.206 + 0.358i)11-s + (0.138 + 0.240i)13-s − 0.258·15-s + (−0.894 − 1.54i)17-s + (−0.0720 − 0.124i)19-s + (−0.367 + 0.637i)21-s + (−0.768 − 1.33i)23-s + 0.200·25-s + 0.192·27-s + (0.127 − 0.220i)29-s + (0.336 − 0.582i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.686 + 0.727i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 0.686 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.474798230\)
\(L(\frac12)\) \(\approx\) \(1.474798230\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
67 \( 1 + (-7.55 - 3.14i)T \)
good7 \( 1 + (1.68 - 2.92i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.686 - 1.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.68 + 6.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.313 + 0.543i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.68 + 6.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.686 + 1.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.87 + 3.24i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.05 - 7.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.31 + 4.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 2.37T + 43T^{2} \)
47 \( 1 + (-3.68 + 6.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.74T + 53T^{2} \)
59 \( 1 - 8.74T + 59T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-2.05 + 3.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.87 + 13.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.05 - 8.76i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.410130952208919475978867749214, −7.75286994552215366755686746092, −6.77778121086037573622897405150, −6.38419985031513196180786307592, −5.24901008893988001661839511301, −4.54374952389755975347980614461, −3.68301470831653715982920561318, −2.54558263575396951309434962677, −2.33962749387423849735500299547, −0.46343611770326132095497620398, 0.950867421841701382995113917762, 2.14049150307294125063812917251, 3.33038481974626290022540654007, 3.83065791111974143272029353699, 4.45748855632819985049046064803, 5.70356549787210982453563559656, 6.42903506909144069838592895909, 7.19033653195552825347473443096, 7.87408375232125650272878088558, 8.387837568644124134953306508013

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