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Properties

Label 2-4020-67.37-c1-0-39
Degree $2$
Conductor $4020$
Sign $-0.978 - 0.205i$
Analytic cond. $32.0998$
Root an. cond. $5.66567$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + (0.5 − 0.866i)7-s + 9-s + (−1.5 + 2.59i)11-s + (−2.5 − 4.33i)13-s − 15-s + (0.5 + 0.866i)17-s + (−3.5 − 6.06i)19-s + (−0.5 + 0.866i)21-s + (4.5 + 7.79i)23-s + 25-s − 27-s + (−2.5 + 4.33i)29-s + (3.5 − 6.06i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + (0.188 − 0.327i)7-s + 0.333·9-s + (−0.452 + 0.783i)11-s + (−0.693 − 1.20i)13-s − 0.258·15-s + (0.121 + 0.210i)17-s + (−0.802 − 1.39i)19-s + (−0.109 + 0.188i)21-s + (0.938 + 1.62i)23-s + 0.200·25-s − 0.192·27-s + (−0.464 + 0.804i)29-s + (0.628 − 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.978 - 0.205i$
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: \(1\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ -0.978 - 0.205i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (8 - 1.73i)T \)
good7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (1.5 - 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (6.5 + 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.75955731592559932021590790032, −7.40813121180611042859697773622, −6.60942653926626933050849191488, −5.70965597585943981153099065223, −5.05617374510400576459452793400, −4.54126959732241335435336408788, −3.30427874521489117648180633922, −2.40873233288986747371158733012, −1.31336201101758474103885204345, 0, 1.49660915590256350690979809263, 2.39023437438900464034270311234, 3.40500919630807755356798885014, 4.63127666657418664780369923765, 4.96006756507958409322549669255, 6.12113148853086871160089205146, 6.34615081915680498219502209388, 7.28449713635753682142262705341, 8.195492169810770934700176949313

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