Properties

Label 2-4020-67.37-c1-0-37
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: $\chi_{4020} (841, \cdot )$
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

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

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