Properties

Label 2-4020-67.37-c1-0-22
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 $0$

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 + (−3.5 − 6.06i)13-s − 15-s + (1.5 + 2.59i)17-s + (2.5 + 4.33i)19-s + (−0.5 + 0.866i)21-s + (1.5 + 2.59i)23-s + 25-s + 27-s + (−0.5 + 0.866i)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 + (−0.970 − 1.68i)13-s − 0.258·15-s + (0.363 + 0.630i)17-s + (0.573 + 0.993i)19-s + (−0.109 + 0.188i)21-s + (0.312 + 0.541i)23-s + 0.200·25-s + 0.192·27-s + (−0.0928 + 0.160i)29-s + (−0.0898 + 0.155i)31-s + ⋯

Functional equation

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

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: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 0.978 + 0.205i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.023280959\)
\(L(\frac12)\) \(\approx\) \(2.023280959\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 6.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.5 + 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-6.5 + 11.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.5 - 9.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.226092511783942496929075882156, −7.77388080323204718278084139125, −7.27369704541936828360470978865, −6.07080960604687865365446256858, −5.51298566459986640142029710113, −4.63128744477594458141505365172, −3.48608425531599299679690638388, −3.16307681379710851525817851492, −2.05005755495356891937845234828, −0.74278018820985881227813629710, 0.841703966956516799376904572713, 2.19695657364563690658098170859, 2.88527970145881767384816990499, 4.00552501085272860820747205290, 4.49590997965160349904598855369, 5.31736138417466182660038346506, 6.61336029544474662903924863331, 7.09321140788080575918386489883, 7.57926087729692550025178697787, 8.523121869379670869261598197449

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