↑

Properties

Label 2-4020-67.37-c1-0-14
Degree $2$
Conductor $4020$
Sign $0.994 + 0.104i$
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 + (−0.775 + 1.34i)7-s + 9-s + (−0.262 + 0.454i)11-s + (−0.0526 − 0.0911i)13-s + 15-s + (1.53 + 2.65i)17-s + (−1.60 − 2.78i)19-s + (0.775 − 1.34i)21-s + (−4.09 − 7.09i)23-s + 25-s − 27-s + (−3.20 + 5.54i)29-s + (1.70 − 2.94i)31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + (−0.292 + 0.507i)7-s + 0.333·9-s + (−0.0791 + 0.137i)11-s + (−0.0145 − 0.0252i)13-s + 0.258·15-s + (0.371 + 0.643i)17-s + (−0.369 − 0.639i)19-s + (0.169 − 0.292i)21-s + (−0.854 − 1.48i)23-s + 0.200·25-s − 0.192·27-s + (−0.594 + 1.03i)29-s + (0.305 − 0.529i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.994 + 0.104i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.994 + 0.104i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.035986827\)
\(L(\frac12)\) \(\approx\) \(1.035986827\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.78 - 2.53i)T \)
good7 \( 1 + (0.775 - 1.34i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.262 - 0.454i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0526 + 0.0911i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.53 - 2.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.60 + 2.78i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.09 + 7.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.20 - 5.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.70 + 2.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.396 + 0.686i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.52 + 4.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4.99T + 43T^{2} \)
47 \( 1 + (-0.703 + 1.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.53T + 53T^{2} \)
59 \( 1 - 3.21T + 59T^{2} \)
61 \( 1 + (-2.81 - 4.87i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-2.92 + 5.06i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.11 + 7.12i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.910 - 1.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.71 - 15.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.71T + 89T^{2} \)
97 \( 1 + (1.03 + 1.79i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.462173927554591270420176046371, −7.66080748484859519812831872285, −6.91469111981671827084602354167, −6.15833832249033290861128392274, −5.58330490362372993479388744454, −4.60661058020126327511251187467, −3.99505536349007832301831971485, −2.91883377076531638461360394524, −1.96367567704209581864646276063, −0.54437470134601700638324161569, 0.64351293895938256388812300070, 1.84277514553982929786201529962, 3.15940979555032168022348414337, 3.90550220513314121358168711397, 4.63352191751330506383670193223, 5.61221881795857461932802926415, 6.14307207187420721616774716341, 7.09350470614177839452210888031, 7.64067220471461602996077111479, 8.268130864866456056077599710991

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