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Properties

Label 2-4020-67.29-c1-0-24
Degree $2$
Conductor $4020$
Sign $-0.153 + 0.988i$
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 + (0.278 + 0.482i)7-s + 9-s + (−0.747 − 1.29i)11-s + (−1.66 + 2.88i)13-s + 15-s + (−3.16 + 5.48i)17-s + (−2.84 + 4.92i)19-s + (−0.278 − 0.482i)21-s + (2.62 − 4.55i)23-s + 25-s − 27-s + (−1.03 − 1.79i)29-s + (−3.14 − 5.45i)31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + (0.105 + 0.182i)7-s + 0.333·9-s + (−0.225 − 0.390i)11-s + (−0.462 + 0.800i)13-s + 0.258·15-s + (−0.768 + 1.33i)17-s + (−0.652 + 1.13i)19-s + (−0.0607 − 0.105i)21-s + (0.548 − 0.949i)23-s + 0.200·25-s − 0.192·27-s + (−0.192 − 0.332i)29-s + (−0.565 − 0.979i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.153 + 0.988i$
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: \((2,\ 4020,\ (\ :1/2),\ -0.153 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4852880395\)
\(L(\frac12)\) \(\approx\) \(0.4852880395\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.44 + 6.87i)T \)
good7 \( 1 + (-0.278 - 0.482i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.747 + 1.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.66 - 2.88i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.16 - 5.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.84 - 4.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.62 + 4.55i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.03 + 1.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.14 + 5.45i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.16 + 7.21i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.52 - 6.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + (-3.75 - 6.50i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.994T + 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 + (1.32 - 2.30i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-1.28 - 2.22i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.29 - 10.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.62 + 8.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.13 - 1.96i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.78T + 89T^{2} \)
97 \( 1 + (-3.17 + 5.49i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.303878167013378803882257136953, −7.50894805427295430191483247469, −6.65498310372778321081853106833, −6.08161539546402679620838765728, −5.33797937073447734390520020379, −4.21193674292178583781878535330, −4.01524684289885735797414898784, −2.56834428924837104585380538354, −1.67410510812476550791816835990, −0.18781854688580996997563184375, 0.892949516534672800057227962585, 2.31231631292664353741471818987, 3.17902478041122660983595477831, 4.24784418934989583067037454382, 5.04629912882070962115911720934, 5.38086531344792647817429298291, 6.74891042702656900627730507045, 7.04079231449122429776062473119, 7.75040998528820329480177955530, 8.691184008970829934716957800808

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