Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.988 + 0.154i$
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 + (2.20 + 0.345i)5-s − 0.0620i·7-s − 9-s + 3.73·11-s − 3.61i·13-s + (−0.345 + 2.20i)15-s − 5.75i·17-s − 4.84·19-s + 0.0620·21-s + 3.34i·23-s + (4.76 + 1.52i)25-s i·27-s − 4.90·29-s + 9.28·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.988 + 0.154i)5-s − 0.0234i·7-s − 0.333·9-s + 1.12·11-s − 1.00i·13-s + (−0.0891 + 0.570i)15-s − 1.39i·17-s − 1.11·19-s + 0.0135·21-s + 0.697i·23-s + (0.952 + 0.305i)25-s − 0.192i·27-s − 0.910·29-s + 1.66·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.988 + 0.154i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ 0.988 + 0.154i)$
$L(1)$  $\approx$  $2.416561974$
$L(\frac12)$  $\approx$  $2.416561974$
$L(\frac{3}{2})$   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 \( 1 + (-2.20 - 0.345i)T \)
67 \( 1 - iT \)
good7 \( 1 + 0.0620iT - 7T^{2} \)
11 \( 1 - 3.73T + 11T^{2} \)
13 \( 1 + 3.61iT - 13T^{2} \)
17 \( 1 + 5.75iT - 17T^{2} \)
19 \( 1 + 4.84T + 19T^{2} \)
23 \( 1 - 3.34iT - 23T^{2} \)
29 \( 1 + 4.90T + 29T^{2} \)
31 \( 1 - 9.28T + 31T^{2} \)
37 \( 1 - 3.35iT - 37T^{2} \)
41 \( 1 - 9.38T + 41T^{2} \)
43 \( 1 + 2.89iT - 43T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 + 9.91iT - 53T^{2} \)
59 \( 1 + 9.42T + 59T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 3.66iT - 83T^{2} \)
89 \( 1 + 5.66T + 89T^{2} \)
97 \( 1 - 2.89iT - 97T^{2} \)
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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

−8.599872744292506802296082685051, −7.72721381180599663533308347687, −6.78617576348971694655592846985, −6.20276568574750059775742351578, −5.41851529003629142194559079706, −4.75956196298751255717801297964, −3.79606364259952775963160079847, −2.93660809426151574227454634413, −2.05782247597768988870067680679, −0.76764151334434437945647322767, 1.14382089998777796317196624649, 1.88747146076460332176074234130, 2.68864475236152672284152887598, 4.07919527263081636049081555679, 4.52029446289201229906857847443, 5.90947113263030741513243421838, 6.22850772220549624960794267559, 6.73638320514817470886906614037, 7.76131797393681556183244443532, 8.565179517846459469002425818160

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