Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.379 + 0.925i$
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 + (−0.847 + 2.06i)5-s + 3.67i·7-s − 9-s − 3.98·11-s − 1.24i·13-s + (2.06 + 0.847i)15-s − 0.354i·17-s − 1.69·19-s + 3.67·21-s − 1.36i·23-s + (−3.56 − 3.50i)25-s + i·27-s + 0.677·29-s + 0.344·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.379 + 0.925i)5-s + 1.38i·7-s − 0.333·9-s − 1.20·11-s − 0.346i·13-s + (0.534 + 0.218i)15-s − 0.0858i·17-s − 0.387·19-s + 0.801·21-s − 0.284i·23-s + (−0.712 − 0.701i)25-s + 0.192i·27-s + 0.125·29-s + 0.0618·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.379 + 0.925i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.379 + 0.925i)$
$L(1)$  $\approx$  $0.3323019677$
$L(\frac12)$  $\approx$  $0.3323019677$
$L(\frac{3}{2})$   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 \( 1 + (0.847 - 2.06i)T \)
67 \( 1 + iT \)
good7 \( 1 - 3.67iT - 7T^{2} \)
11 \( 1 + 3.98T + 11T^{2} \)
13 \( 1 + 1.24iT - 13T^{2} \)
17 \( 1 + 0.354iT - 17T^{2} \)
19 \( 1 + 1.69T + 19T^{2} \)
23 \( 1 + 1.36iT - 23T^{2} \)
29 \( 1 - 0.677T + 29T^{2} \)
31 \( 1 - 0.344T + 31T^{2} \)
37 \( 1 - 8.16iT - 37T^{2} \)
41 \( 1 + 5.45T + 41T^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 - 10.1iT - 47T^{2} \)
53 \( 1 - 4.74iT - 53T^{2} \)
59 \( 1 - 5.13T + 59T^{2} \)
61 \( 1 - 5.19T + 61T^{2} \)
71 \( 1 + 3.56T + 71T^{2} \)
73 \( 1 + 4.89iT - 73T^{2} \)
79 \( 1 - 5.19T + 79T^{2} \)
83 \( 1 + 7.49iT - 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + 17.6iT - 97T^{2} \)
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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

−8.305565595554742800596317513832, −7.47180590948857050130842186422, −6.80092031827859918570551505598, −5.97940113051864612984352919684, −5.46125972702817438234072336723, −4.50266433654314162932632246816, −3.15407530465865373570542536036, −2.72042003759165761755195923228, −1.90556505200146854873133157275, −0.10781105551788223565013949916, 0.975545527366455951186270920974, 2.29024444027464271285965698113, 3.60285278580153468342001183701, 4.06191407199640111116251811881, 4.91501609699515467323453733715, 5.38994462922118434430296728833, 6.53301431895511255883972703384, 7.38175037094347775446030853986, 7.984367615442820482389753760828, 8.584770133538296702106958632884

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