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Properties

Label 2-4020-201.200-c1-0-23
Degree $2$
Conductor $4020$
Sign $0.882 - 0.471i$
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  + (0.370 − 1.69i)3-s + 5-s + 0.0825i·7-s + (−2.72 − 1.25i)9-s − 1.67·11-s + 4.50i·13-s + (0.370 − 1.69i)15-s + 1.74i·17-s − 1.02·19-s + (0.139 + 0.0305i)21-s + 3.31i·23-s + 25-s + (−3.13 + 4.14i)27-s − 3.47i·29-s − 4.73i·31-s + ⋯
L(s)  = 1  + (0.213 − 0.976i)3-s + 0.447·5-s + 0.0312i·7-s + (−0.908 − 0.417i)9-s − 0.503·11-s + 1.24i·13-s + (0.0956 − 0.436i)15-s + 0.422i·17-s − 0.234·19-s + (0.0304 + 0.00667i)21-s + 0.692i·23-s + 0.200·25-s + (−0.602 + 0.798i)27-s − 0.645i·29-s − 0.850i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.882 - 0.471i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 0.882 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.657379984\)
\(L(\frac12)\) \(\approx\) \(1.657379984\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.370 + 1.69i)T \)
5 \( 1 - T \)
67 \( 1 + (5.31 + 6.22i)T \)
good7 \( 1 - 0.0825iT - 7T^{2} \)
11 \( 1 + 1.67T + 11T^{2} \)
13 \( 1 - 4.50iT - 13T^{2} \)
17 \( 1 - 1.74iT - 17T^{2} \)
19 \( 1 + 1.02T + 19T^{2} \)
23 \( 1 - 3.31iT - 23T^{2} \)
29 \( 1 + 3.47iT - 29T^{2} \)
31 \( 1 + 4.73iT - 31T^{2} \)
37 \( 1 - 1.03T + 37T^{2} \)
41 \( 1 - 4.96T + 41T^{2} \)
43 \( 1 - 6.28iT - 43T^{2} \)
47 \( 1 - 12.7iT - 47T^{2} \)
53 \( 1 + 1.52T + 53T^{2} \)
59 \( 1 - 7.15iT - 59T^{2} \)
61 \( 1 - 1.24iT - 61T^{2} \)
71 \( 1 - 8.32iT - 71T^{2} \)
73 \( 1 - 9.54T + 73T^{2} \)
79 \( 1 - 4.58iT - 79T^{2} \)
83 \( 1 - 2.48iT - 83T^{2} \)
89 \( 1 - 2.00iT - 89T^{2} \)
97 \( 1 + 8.09iT - 97T^{2} \)
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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

−8.405960334600936557267800461626, −7.69448922390273510461787416761, −7.13805820863841706932467087660, −6.13402606008174534296193323453, −5.94830905103764556004471098857, −4.75290939308258643680188353127, −3.87386661506927251939015546616, −2.71622525589024582409313035693, −2.07023899256166429551575837441, −1.10447241474000083808396902225, 0.48759801156173205395921222125, 2.15842652302389622022901336599, 2.96757103761359528306620890460, 3.68751624740381594991323648236, 4.76616775296581480189275467579, 5.28977805254941460141059511683, 5.94047870679524040188232522532, 6.94488784604689906374475653927, 7.81478650166990427165150404478, 8.541984539177977882460219351977

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