Properties

Label 2-4020-67.37-c1-0-38
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: $\chi_{4020} (841, \cdot )$
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

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

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