Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $-0.566 - 0.824i$
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 + (−1.13 − 1.97i)7-s + 9-s + (1.34 + 2.32i)11-s + (−0.0777 + 0.134i)13-s − 15-s + (−0.386 + 0.669i)17-s + (−0.333 + 0.576i)19-s + (1.13 + 1.97i)21-s + (−2.24 + 3.88i)23-s + 25-s − 27-s + (0.790 + 1.36i)29-s + (−2.92 − 5.06i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + (−0.430 − 0.745i)7-s + 0.333·9-s + (0.404 + 0.700i)11-s + (−0.0215 + 0.0373i)13-s − 0.258·15-s + (−0.0937 + 0.162i)17-s + (−0.0764 + 0.132i)19-s + (0.248 + 0.430i)21-s + (−0.467 + 0.809i)23-s + 0.200·25-s − 0.192·27-s + (0.146 + 0.254i)29-s + (−0.525 − 0.909i)31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-0.566 - 0.824i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (3781, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ -0.566 - 0.824i)$
$L(1)$  $\approx$  $0.6399756355$
$L(\frac12)$  $\approx$  $0.6399756355$
$L(\frac{3}{2})$   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 \( 1 + (1.48 + 8.04i)T \)
good7 \( 1 + (1.13 + 1.97i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.34 - 2.32i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.0777 - 0.134i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.386 - 0.669i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.333 - 0.576i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.24 - 3.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.790 - 1.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.92 + 5.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.60 - 7.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.88 + 8.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4.88T + 43T^{2} \)
47 \( 1 + (-4.20 - 7.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.78T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 + (-1.14 + 1.98i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-6.37 - 11.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.73 - 6.47i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.85 + 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.19 - 7.26i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.03T + 89T^{2} \)
97 \( 1 + (5.24 - 9.08i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.805680373281029043315124402840, −7.79402065823744273132850836259, −7.08444070283385028979621020759, −6.56321160186949905293736688905, −5.77169036291833640528898631687, −5.01229922272957985083299011869, −4.13555402103027563057948026337, −3.45826360972805800456698001748, −2.13612479685547947509275836944, −1.22319116109044082862061118669, 0.20383383770134357620869023909, 1.57213592032214889268937155692, 2.60834009416157604730123488888, 3.50092153464342225633422567759, 4.50930742038190656596547699138, 5.40516868967426238721006592704, 5.93547745365513215060521601968, 6.58869487715481073077547776178, 7.25969173113803916718578990742, 8.433868081718692736292133287423

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