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Properties

Label 2-4020-67.29-c1-0-1
Degree $2$
Conductor $4020$
Sign $-0.999 + 0.00463i$
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 + (2.11 + 3.66i)7-s + 9-s + (−1.59 − 2.76i)11-s + (−3.28 + 5.69i)13-s + 15-s + (3.94 − 6.82i)17-s + (−0.787 + 1.36i)19-s + (−2.11 − 3.66i)21-s + (−1.83 + 3.18i)23-s + 25-s − 27-s + (3.82 + 6.63i)29-s + (−1.14 − 1.98i)31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + (0.799 + 1.38i)7-s + 0.333·9-s + (−0.481 − 0.833i)11-s + (−0.912 + 1.58i)13-s + 0.258·15-s + (0.955 − 1.65i)17-s + (−0.180 + 0.312i)19-s + (−0.461 − 0.799i)21-s + (−0.383 + 0.664i)23-s + 0.200·25-s − 0.192·27-s + (0.711 + 1.23i)29-s + (−0.205 − 0.355i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.999 + 0.00463i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (3781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ -0.999 + 0.00463i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4780101908\)
\(L(\frac12)\) \(\approx\) \(0.4780101908\)
\(L(\frac{3}{2})\) 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 \( 1 + (-7.48 - 3.30i)T \)
good7 \( 1 + (-2.11 - 3.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.59 + 2.76i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.28 - 5.69i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.94 + 6.82i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.787 - 1.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.83 - 3.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.82 - 6.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.14 + 1.98i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.16 - 8.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.81 + 4.87i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 0.735T + 43T^{2} \)
47 \( 1 + (-2.26 - 3.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.70T + 53T^{2} \)
59 \( 1 - 4.74T + 59T^{2} \)
61 \( 1 + (-4.83 + 8.37i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (3.13 + 5.43i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.15 - 8.93i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.63 + 8.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.0412 + 0.0715i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.35T + 89T^{2} \)
97 \( 1 + (3.08 - 5.34i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.762180997695427772968281102690, −8.125985939804490744090654738575, −7.30540988082631039227099304429, −6.66826761773785944078399200417, −5.60245333041589105321241511975, −5.16856305610258600703714003059, −4.56327840409016446342029799168, −3.32310217642235500972660145389, −2.45194110801216139094840319604, −1.42577904331023964416541539573, 0.16146241043767646055600857929, 1.19213476836498203409889443987, 2.39561849604654883259124104474, 3.67063184197761735784960170031, 4.30827201142085910327689640518, 5.03405509925087961536520018578, 5.71605532243111183344023250513, 6.75862615534310106206840031380, 7.51126373199810254503442362859, 7.87846220462004422902887515364

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