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Properties

Label 2-4020-67.29-c1-0-45
Degree $2$
Conductor $4020$
Sign $-0.860 + 0.509i$
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 + (−2.09 − 3.62i)7-s + 9-s + (−1.91 − 3.32i)11-s + (3.12 − 5.41i)13-s + 15-s + (−0.293 + 0.508i)17-s + (−1.03 + 1.78i)19-s + (−2.09 − 3.62i)21-s + (−0.208 + 0.360i)23-s + 25-s + 27-s + (−2.56 − 4.43i)29-s + (0.705 + 1.22i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + (−0.790 − 1.36i)7-s + 0.333·9-s + (−0.578 − 1.00i)11-s + (0.866 − 1.50i)13-s + 0.258·15-s + (−0.0712 + 0.123i)17-s + (−0.236 + 0.409i)19-s + (−0.456 − 0.790i)21-s + (−0.0434 + 0.0751i)23-s + 0.200·25-s + 0.192·27-s + (−0.475 − 0.823i)29-s + (0.126 + 0.219i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.860 + 0.509i$
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: \((2,\ 4020,\ (\ :1/2),\ -0.860 + 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.552144258\)
\(L(\frac12)\) \(\approx\) \(1.552144258\)
\(L(\frac{3}{2})\) 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 \( 1 + (-8.13 + 0.936i)T \)
good7 \( 1 + (2.09 + 3.62i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.91 + 3.32i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.12 + 5.41i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.293 - 0.508i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.03 - 1.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.208 - 0.360i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.56 + 4.43i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.705 - 1.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.95 - 6.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.442 + 0.766i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4.05T + 43T^{2} \)
47 \( 1 + (-0.533 - 0.923i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.55T + 53T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 + (2.87 - 4.98i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (7.60 + 13.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.89 - 10.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.23 + 3.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.17 - 2.03i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.94T + 89T^{2} \)
97 \( 1 + (3.07 - 5.32i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.137175809491886883989540229381, −7.57093061366329648733619696312, −6.62479094963331501149029906324, −6.02247254083225582252675015682, −5.23529417478803129416656367202, −4.06538103980287823632555833803, −3.38283485434430980931575664755, −2.85673982296275984742654546793, −1.40944051544625993634211205033, −0.38813601048778277540884227200, 1.76545210617723392952948960036, 2.28770106150961661033001932728, 3.18383891996836599248796884077, 4.14366899897317377078903136487, 5.05458289601253609294468371052, 5.82427534743702442292883232433, 6.65466387958527682278381926476, 7.10068788583720837145820532856, 8.228116931120811582414899178361, 8.940091794802973237414144238229

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