Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.905 - 0.423i$
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 + (−2.02 − 0.947i)5-s + 2.95i·7-s − 9-s + 2.74·11-s − 5.48i·13-s + (−0.947 + 2.02i)15-s − 0.836i·17-s − 7.78·19-s + 2.95·21-s − 1.14i·23-s + (3.20 + 3.83i)25-s + i·27-s + 3.28·29-s + 5.40·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.905 − 0.423i)5-s + 1.11i·7-s − 0.333·9-s + 0.828·11-s − 1.52i·13-s + (−0.244 + 0.522i)15-s − 0.202i·17-s − 1.78·19-s + 0.645·21-s − 0.238i·23-s + (0.641 + 0.767i)25-s + 0.192i·27-s + 0.610·29-s + 0.971·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.905 - 0.423i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (1609, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.905 - 0.423i)$
$L(1)$  $\approx$  $0.1874201078$
$L(\frac12)$  $\approx$  $0.1874201078$
$L(\frac{3}{2})$   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 \( 1 + (2.02 + 0.947i)T \)
67 \( 1 + iT \)
good7 \( 1 - 2.95iT - 7T^{2} \)
11 \( 1 - 2.74T + 11T^{2} \)
13 \( 1 + 5.48iT - 13T^{2} \)
17 \( 1 + 0.836iT - 17T^{2} \)
19 \( 1 + 7.78T + 19T^{2} \)
23 \( 1 + 1.14iT - 23T^{2} \)
29 \( 1 - 3.28T + 29T^{2} \)
31 \( 1 - 5.40T + 31T^{2} \)
37 \( 1 - 4.63iT - 37T^{2} \)
41 \( 1 - 1.97T + 41T^{2} \)
43 \( 1 + 5.74iT - 43T^{2} \)
47 \( 1 + 2.67iT - 47T^{2} \)
53 \( 1 - 3.93iT - 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 8.93T + 61T^{2} \)
71 \( 1 + 5.94T + 71T^{2} \)
73 \( 1 - 0.964iT - 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 9.18iT - 83T^{2} \)
89 \( 1 - 8.47T + 89T^{2} \)
97 \( 1 + 2.98iT - 97T^{2} \)
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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

−8.135532127579312462252018474979, −7.43485387021776218255952150169, −6.44329815093293790053696197027, −5.96808370390372183877781198926, −5.00262368001193259682283428550, −4.27812971803077989743900430229, −3.20405799649315734242480735155, −2.47563808036214069962191345029, −1.23986894081123625877999942958, −0.05816932761494927283519681300, 1.41761513223210780608805729798, 2.71232057341424618787979338921, 3.82993752696655131998786863782, 4.23648284332541877523702365926, 4.63924336691308694752888049497, 6.26245826211452372051999870110, 6.58774352261982548884043143708, 7.36450571402889180383939769971, 8.147236000662659523827067985414, 8.868500901826939493476572335943

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