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Properties

Label 2-4020-67.37-c1-0-12
Degree $2$
Conductor $4020$
Sign $0.604 - 0.796i$
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 + (2.26 − 3.93i)7-s + 9-s + (−2.47 + 4.29i)11-s + (0.5 + 0.866i)13-s − 15-s + (0.378 + 0.655i)17-s + (2.83 + 4.91i)19-s + (2.26 − 3.93i)21-s + (4.02 + 6.97i)23-s + 25-s + 27-s + (0.975 − 1.68i)29-s + (−4.04 + 7.00i)31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + (0.857 − 1.48i)7-s + 0.333·9-s + (−0.747 + 1.29i)11-s + (0.138 + 0.240i)13-s − 0.258·15-s + (0.0917 + 0.158i)17-s + (0.650 + 1.12i)19-s + (0.495 − 0.857i)21-s + (0.839 + 1.45i)23-s + 0.200·25-s + 0.192·27-s + (0.181 − 0.313i)29-s + (−0.726 + 1.25i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.604 - 0.796i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.604 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.107221454\)
\(L(\frac12)\) \(\approx\) \(2.107221454\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.85 + 7.97i)T \)
good7 \( 1 + (-2.26 + 3.93i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.47 - 4.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.378 - 0.655i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.83 - 4.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.02 - 6.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.975 + 1.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.04 - 7.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.35 + 4.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.516 + 0.894i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 8.67T + 43T^{2} \)
47 \( 1 + (5.48 - 9.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.08T + 53T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 + (1.14 + 1.97i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (2.10 - 3.64i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.29 - 12.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.59 + 9.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.15 - 8.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + (-8.53 - 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.242911492797285842861563424478, −7.77228234110688845251785527592, −7.32567219686772918948889179429, −6.70778636764898983407165549769, −5.22631033947632034581203241474, −4.80369503695019588294341464317, −3.83062951101907419594697000322, −3.36871197504548254049267355466, −1.92029377951126167351488275453, −1.23126512197581024446080974818, 0.58806087892235202257402338256, 2.05604471370584324771719207221, 2.85002356710524738873199029230, 3.42145216742297933634137499977, 4.83796608510809281096670436663, 5.14629283757012996829889902564, 6.06097991708626172142532350461, 6.95954443904734162842386257239, 7.892921374852636615237259964630, 8.493105201137375646663388564672

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