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Properties

Label 2-4020-67.37-c1-0-25
Degree $2$
Conductor $4020$
Sign $0.856 + 0.516i$
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 + (−0.550 + 0.952i)7-s + 9-s + (−0.589 + 1.02i)11-s + (0.5 + 0.866i)13-s − 15-s + (−1.83 − 3.17i)17-s + (−3.26 − 5.65i)19-s + (−0.550 + 0.952i)21-s + (4.11 + 7.12i)23-s + 25-s + 27-s + (4.58 − 7.94i)29-s + (1.12 − 1.94i)31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + (−0.207 + 0.360i)7-s + 0.333·9-s + (−0.177 + 0.307i)11-s + (0.138 + 0.240i)13-s − 0.258·15-s + (−0.443 − 0.768i)17-s + (−0.748 − 1.29i)19-s + (−0.120 + 0.207i)21-s + (0.857 + 1.48i)23-s + 0.200·25-s + 0.192·27-s + (0.851 − 1.47i)29-s + (0.201 − 0.349i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.856 + 0.516i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.856 + 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.894603304\)
\(L(\frac12)\) \(\approx\) \(1.894603304\)
\(L(\frac{3}{2})\) 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 \( 1 + (-8.12 - 0.997i)T \)
good7 \( 1 + (0.550 - 0.952i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.589 - 1.02i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.83 + 3.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.26 + 5.65i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.11 - 7.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.58 + 7.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.12 + 1.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.58 + 9.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.73 - 8.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 3.52T + 43T^{2} \)
47 \( 1 + (-5.16 + 8.94i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.24T + 53T^{2} \)
59 \( 1 - 8.73T + 59T^{2} \)
61 \( 1 + (-7.40 - 12.8i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-3.07 + 5.32i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.45 + 7.71i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.22 - 2.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.48 + 9.50i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.19T + 89T^{2} \)
97 \( 1 + (-3.18 - 5.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.612194878123555016451889023891, −7.48664059975276917486052336011, −7.16362327262587199711070115987, −6.28793663674368628169565124491, −5.28535943989077789326053097964, −4.52409239005001139306440205763, −3.76591613120541798150323915225, −2.76841927859448179166847158252, −2.14783630828241307783737962103, −0.62368182913077868489132600036, 0.935880450309957144400092918412, 2.13440519239245292889334571323, 3.16910718890180217389467494963, 3.79883333888249978928818486251, 4.59776957551658394308941262571, 5.48203199004891723013568625345, 6.64976927100535310846253657774, 6.85881055671282205890082226719, 8.035616814332804518646415596293, 8.492713489702654093540207135726

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