Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.0649 - 0.997i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (0.145 − 2.23i)5-s + 1.12i·7-s − 9-s + 6.26·11-s + 0.929i·13-s + (2.23 + 0.145i)15-s + 1.84i·17-s − 5.56·19-s − 1.12·21-s + 6.60i·23-s + (−4.95 − 0.648i)25-s i·27-s + 2.69·29-s − 10.1·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.0649 − 0.997i)5-s + 0.426i·7-s − 0.333·9-s + 1.88·11-s + 0.257i·13-s + (0.576 + 0.0375i)15-s + 0.448i·17-s − 1.27·19-s − 0.246·21-s + 1.37i·23-s + (−0.991 − 0.129i)25-s − 0.192i·27-s + 0.501·29-s − 1.83·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0649 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0649 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.0649 - 0.997i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ 0.0649 - 0.997i)$
$L(1)$  $\approx$  $1.689352509$
$L(\frac12)$  $\approx$  $1.689352509$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;67\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - iT \)
5 \( 1 + (-0.145 + 2.23i)T \)
67 \( 1 - iT \)
good7 \( 1 - 1.12iT - 7T^{2} \)
11 \( 1 - 6.26T + 11T^{2} \)
13 \( 1 - 0.929iT - 13T^{2} \)
17 \( 1 - 1.84iT - 17T^{2} \)
19 \( 1 + 5.56T + 19T^{2} \)
23 \( 1 - 6.60iT - 23T^{2} \)
29 \( 1 - 2.69T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 - 3.96iT - 37T^{2} \)
41 \( 1 - 6.94T + 41T^{2} \)
43 \( 1 - 8.50iT - 43T^{2} \)
47 \( 1 + 2.02iT - 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
71 \( 1 - 1.79T + 71T^{2} \)
73 \( 1 + 8.93iT - 73T^{2} \)
79 \( 1 + 0.0306T + 79T^{2} \)
83 \( 1 + 14.1iT - 83T^{2} \)
89 \( 1 + 5.95T + 89T^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.913023258489198253698643958033, −8.105217733904407774481968456184, −7.14920841087113717229204275596, −6.13969927400928442063844014631, −5.77630932153697476506479149048, −4.67819873764613003027676508589, −4.14333426372333743301998313000, −3.46034964928915219623945470566, −2.02442080563132611565905977334, −1.21049929776876906850507808862, 0.50971871448712885895843055566, 1.80692042644618330394418501525, 2.59289612723225235965896677005, 3.76175405290272973342687915851, 4.15764780564759615660585368303, 5.50343908133361091339492146059, 6.34417171076290344487413829191, 6.83441440114534372576962728257, 7.24373779183726650267975061542, 8.269143398470735138371544503754

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