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Properties

Label 2-4020-67.37-c1-0-45
Degree $2$
Conductor $4020$
Sign $-0.978 - 0.205i$
Analytic cond. $32.0998$
Root an. cond. $5.66567$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + (0.5 − 0.866i)7-s + 9-s + (−1.5 + 2.59i)11-s + (−2.5 − 4.33i)13-s − 15-s + (−1.5 − 2.59i)17-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)21-s + (−1.5 − 2.59i)23-s + 25-s + 27-s + (1.5 − 2.59i)29-s + (−2.5 + 4.33i)31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + (0.188 − 0.327i)7-s + 0.333·9-s + (−0.452 + 0.783i)11-s + (−0.693 − 1.20i)13-s − 0.258·15-s + (−0.363 − 0.630i)17-s + (0.114 + 0.198i)19-s + (0.109 − 0.188i)21-s + (−0.312 − 0.541i)23-s + 0.200·25-s + 0.192·27-s + (0.278 − 0.482i)29-s + (−0.449 + 0.777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.978 - 0.205i$
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: \(1\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ -0.978 - 0.205i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (8 - 1.73i)T \)
good7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (1.5 - 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.5 + 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.067172311913799644398522518899, −7.40808690279722983292084707155, −6.87688047559798852466584787865, −5.77833326632509560543595591078, −4.76420415923392057797651200723, −4.42520992245717355234626174129, −3.15904109307615042658096502722, −2.67081687129210301770040154055, −1.43006330164588165222317541853, 0, 1.66762510054185849584593109074, 2.48700542010702159211854081058, 3.46584461151188074148618958531, 4.19217596240283791760325536946, 5.01419559473379600833467184140, 5.89667737031181707541371212422, 6.74536946460797627791957320597, 7.53609979189000858449249496835, 8.053009398975252598951507919156

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