Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.995 - 0.0987i$
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 + (1.26 + 2.19i)7-s + 9-s + (0.0636 + 0.110i)11-s + (−2.22 + 3.85i)13-s − 15-s + (0.129 − 0.224i)17-s + (3.21 − 5.56i)19-s + (−1.26 − 2.19i)21-s + (3.09 − 5.35i)23-s + 25-s − 27-s + (−0.920 − 1.59i)29-s + (−2.61 − 4.53i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + (0.477 + 0.827i)7-s + 0.333·9-s + (0.0191 + 0.0332i)11-s + (−0.617 + 1.06i)13-s − 0.258·15-s + (0.0314 − 0.0544i)17-s + (0.737 − 1.27i)19-s + (−0.275 − 0.477i)21-s + (0.644 − 1.11i)23-s + 0.200·25-s − 0.192·27-s + (−0.170 − 0.295i)29-s + (−0.470 − 0.814i)31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0987i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0987i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.995 - 0.0987i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (3781, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ 0.995 - 0.0987i)$
$L(1)$  $\approx$  $1.795566728$
$L(\frac12)$  $\approx$  $1.795566728$
$L(\frac{3}{2})$   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 \( 1 + (-7.76 - 2.58i)T \)
good7 \( 1 + (-1.26 - 2.19i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0636 - 0.110i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.22 - 3.85i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.129 + 0.224i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.21 + 5.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.09 + 5.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.920 + 1.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.61 + 4.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.92 + 8.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.579 - 1.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.35T + 43T^{2} \)
47 \( 1 + (-5.48 - 9.50i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.37T + 53T^{2} \)
59 \( 1 + 0.400T + 59T^{2} \)
61 \( 1 + (-2.78 + 4.82i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-3.62 - 6.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.00 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.753 - 1.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.21 - 5.56i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.27T + 89T^{2} \)
97 \( 1 + (2.93 - 5.07i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.586308976081480817261454847355, −7.57685666143692469744246793901, −6.95039009487723144468199974959, −6.19412230379722805575507074288, −5.44496476145265193277606958354, −4.82384847499998091366796477935, −4.11236819556748116156718700219, −2.64167839352266048280832092520, −2.12594034638440251200920064983, −0.77069607701553889850816002746, 0.862726364644656957896429755955, 1.69416362419000046442642041230, 3.04658105307728561637809726095, 3.82860448744456961763419225943, 4.90537921717008944545923058560, 5.39789081791328154059068532794, 6.08841712735860375576260038965, 7.13582830800905872009550582626, 7.54467460085873622295472294262, 8.284301497392635546033634636980

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