Properties

Label 2-4020-5.4-c1-0-60
Degree $2$
Conductor $4020$
Sign $-0.997 + 0.0727i$
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  i·3-s + (2.23 − 0.162i)5-s − 0.770i·7-s − 9-s − 0.920·11-s − 3.41i·13-s + (−0.162 − 2.23i)15-s − 2.39i·17-s − 5.29·19-s − 0.770·21-s − 3.81i·23-s + (4.94 − 0.725i)25-s + i·27-s − 3.65·29-s − 10.5·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.997 − 0.0727i)5-s − 0.291i·7-s − 0.333·9-s − 0.277·11-s − 0.947i·13-s + (−0.0420 − 0.575i)15-s − 0.579i·17-s − 1.21·19-s − 0.168·21-s − 0.794i·23-s + (0.989 − 0.145i)25-s + 0.192i·27-s − 0.678·29-s − 1.90·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.997 + 0.0727i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (1609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ -0.997 + 0.0727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.012113800\)
\(L(\frac12)\) \(\approx\) \(1.012113800\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (-2.23 + 0.162i)T \)
67 \( 1 - iT \)
good7 \( 1 + 0.770iT - 7T^{2} \)
11 \( 1 + 0.920T + 11T^{2} \)
13 \( 1 + 3.41iT - 13T^{2} \)
17 \( 1 + 2.39iT - 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 + 3.81iT - 23T^{2} \)
29 \( 1 + 3.65T + 29T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 + 0.493iT - 37T^{2} \)
41 \( 1 + 6.61T + 41T^{2} \)
43 \( 1 - 11.1iT - 43T^{2} \)
47 \( 1 + 4.33iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 2.20T + 59T^{2} \)
61 \( 1 - 1.58T + 61T^{2} \)
71 \( 1 + 3.39T + 71T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 - 0.553T + 79T^{2} \)
83 \( 1 + 7.19iT - 83T^{2} \)
89 \( 1 + 2.49T + 89T^{2} \)
97 \( 1 + 9.99iT - 97T^{2} \)
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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

−8.037422315442803243266735946265, −7.32470002731610392265767869553, −6.60805350381780219032401299274, −5.87101677640391375755605523317, −5.28932947445927361545511181004, −4.37679138552702288799971689702, −3.19211930818630957880910296271, −2.38383215183536282160329521192, −1.52488397123073299837502154410, −0.25741419141062363363904790640, 1.73167351248020382391816274290, 2.27529583091893708012520295004, 3.51555383654871142937881225035, 4.19434850262298846730127057680, 5.29822897635047315667109727742, 5.63795006296362856227528124454, 6.56728702044008037637014748946, 7.17803619565334941320926048947, 8.316958761371476383530161929184, 8.944276284442354764118282742206

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