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Properties

Label 2-4020-67.37-c1-0-28
Degree $2$
Conductor $4020$
Sign $0.806 + 0.591i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + (0.5 − 0.866i)7-s + 9-s + (0.5 − 0.866i)11-s + (1.5 + 2.59i)13-s − 15-s + (2.5 + 4.33i)17-s + (−3.5 − 6.06i)19-s + (−0.5 + 0.866i)21-s + (−1.5 − 2.59i)23-s + 25-s − 27-s + (3.5 − 6.06i)29-s + (−0.5 + 0.866i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + (0.188 − 0.327i)7-s + 0.333·9-s + (0.150 − 0.261i)11-s + (0.416 + 0.720i)13-s − 0.258·15-s + (0.606 + 1.05i)17-s + (−0.802 − 1.39i)19-s + (−0.109 + 0.188i)21-s + (−0.312 − 0.541i)23-s + 0.200·25-s − 0.192·27-s + (0.649 − 1.12i)29-s + (−0.0898 + 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.806 + 0.591i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.806 + 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.729533984\)
\(L(\frac12)\) \(\approx\) \(1.729533984\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (7.5 - 12.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.5 + 9.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
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

−8.494910501804390476884110801865, −7.56806846076379741500717265976, −6.70474618580469433972554903946, −6.21186672293638250179773881293, −5.52881154013073018092948689160, −4.45562296801687955525646812023, −4.06821239711296237091962497883, −2.75613455341640813542036081651, −1.76095255939046049382677383420, −0.67558602119566158094068470065, 0.955192796212620988039633999473, 1.95561095729193591857360997127, 3.03962634947120683570838337488, 3.98416875751325729197461905429, 4.93755883565379582843207128780, 5.65741416593099455815615141533, 6.08711984746274730597246522241, 7.04122317915188098145382334114, 7.74890340268782608617751992814, 8.497567595182442472127483938543

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