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Properties

Label 2-4020-67.29-c1-0-17
Degree $2$
Conductor $4020$
Sign $-0.0297 - 0.999i$
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 + (2.37 + 4.12i)7-s + 9-s + (2.09 + 3.62i)11-s + (0.5 − 0.866i)13-s − 15-s + (−1.68 + 2.91i)17-s + (2.14 − 3.72i)19-s + (2.37 + 4.12i)21-s + (−0.737 + 1.27i)23-s + 25-s + 27-s + (2.87 + 4.97i)29-s + (1.32 + 2.28i)31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + (0.899 + 1.55i)7-s + 0.333·9-s + (0.630 + 1.09i)11-s + (0.138 − 0.240i)13-s − 0.258·15-s + (−0.408 + 0.706i)17-s + (0.493 − 0.854i)19-s + (0.519 + 0.899i)21-s + (−0.153 + 0.266i)23-s + 0.200·25-s + 0.192·27-s + (0.533 + 0.923i)29-s + (0.237 + 0.410i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0297 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0297 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.0297 - 0.999i$
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: \((2,\ 4020,\ (\ :1/2),\ -0.0297 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.467852632\)
\(L(\frac12)\) \(\approx\) \(2.467852632\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.11 + 7.56i)T \)
good7 \( 1 + (-2.37 - 4.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.09 - 3.62i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.68 - 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.14 + 3.72i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.737 - 1.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.87 - 4.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.32 - 2.28i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.02 + 8.70i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0825 + 0.142i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 7.29T + 43T^{2} \)
47 \( 1 + (-2.73 - 4.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.64T + 53T^{2} \)
59 \( 1 + 8.85T + 59T^{2} \)
61 \( 1 + (-3.86 + 6.69i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-6.87 - 11.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.88 - 11.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.08 + 3.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.24 + 12.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.17T + 89T^{2} \)
97 \( 1 + (7.67 - 13.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.614981988452147230506360767654, −8.062495861688662961121407063520, −7.27258840567447120846224677349, −6.52978104449552009954033486909, −5.54247397800466640590347692388, −4.82619505381100641399686756459, −4.13764492504021578202838587508, −3.05180975920605591118550028454, −2.23073447934802708067028940135, −1.43381447325651215886322517179, 0.69511195426649139188586618242, 1.51340084635741512286458989984, 2.86674646764445614211635374861, 3.78353586384031820289053165987, 4.24304273016593289947701694015, 5.03641318934801080423944283824, 6.25319429475898073075614846103, 6.89818641115996101230493725967, 7.74199777841494168655497627670, 8.123443105236397928313315531172

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