Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.712 - 0.701i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + (2.01 + 3.48i)7-s + 9-s + (1.47 + 2.56i)11-s + (−0.795 + 1.37i)13-s − 15-s + (−2.62 + 4.54i)17-s + (0.339 − 0.587i)19-s + (−2.01 − 3.48i)21-s + (−0.358 + 0.620i)23-s + 25-s − 27-s + (3.40 + 5.90i)29-s + (1.61 + 2.78i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + (0.761 + 1.31i)7-s + 0.333·9-s + (0.445 + 0.772i)11-s + (−0.220 + 0.382i)13-s − 0.258·15-s + (−0.636 + 1.10i)17-s + (0.0778 − 0.134i)19-s + (−0.439 − 0.761i)21-s + (−0.0747 + 0.129i)23-s + 0.200·25-s − 0.192·27-s + (0.632 + 1.09i)29-s + (0.289 + 0.500i)31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.712 - 0.701i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (3781, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.712 - 0.701i)$
$L(1)$  $\approx$  $1.459353467$
$L(\frac12)$  $\approx$  $1.459353467$
$L(\frac{3}{2})$   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 \( 1 + T \)
5 \( 1 - T \)
67 \( 1 + (2.98 + 7.62i)T \)
good7 \( 1 + (-2.01 - 3.48i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.47 - 2.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.795 - 1.37i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.62 - 4.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.339 + 0.587i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.358 - 0.620i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.40 - 5.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.61 - 2.78i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.94 - 3.37i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.662 + 1.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (3.03 + 5.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.84T + 53T^{2} \)
59 \( 1 - 3.21T + 59T^{2} \)
61 \( 1 + (-0.514 + 0.890i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (6.97 + 12.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.44 - 5.96i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.19 - 8.99i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.85 + 6.68i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.969T + 89T^{2} \)
97 \( 1 + (-4.77 + 8.26i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.681411222976241486491317292243, −8.202106758925625268653405659956, −6.98876927913309970679838252738, −6.53827590270968056441955744012, −5.70656305119238006440862464690, −4.99334629812139658903699287869, −4.49365254336365221611431008645, −3.24059251767992937219882749184, −2.00901533297073767895410798555, −1.59656304348831158282694346180, 0.46098479019010255791456453202, 1.31071427810578543167039692682, 2.53283519898041596586116145594, 3.67707563360051303883159501388, 4.50743720478396928103303738612, 5.07748838870412033134113451350, 6.00922297924531878680213772049, 6.68441561718860285307863919444, 7.38357148110055798640959838045, 8.069544045829136408953991076363

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