Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.444 + 0.895i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.885 − 1.10i)2-s + (−0.430 + 1.95i)4-s + 3.50i·5-s − 7-s + (2.53 − 1.25i)8-s + (3.85 − 3.10i)10-s − 3.01i·11-s − 3.90i·13-s + (0.885 + 1.10i)14-s + (−3.62 − 1.68i)16-s − 1.38·17-s − 4.79i·19-s + (−6.83 − 1.50i)20-s + (−3.32 + 2.66i)22-s + 5.06·23-s + ⋯
L(s)  = 1  + (−0.626 − 0.779i)2-s + (−0.215 + 0.976i)4-s + 1.56i·5-s − 0.377·7-s + (0.895 − 0.444i)8-s + (1.22 − 0.980i)10-s − 0.908i·11-s − 1.08i·13-s + (0.236 + 0.294i)14-s + (−0.907 − 0.420i)16-s − 0.336·17-s − 1.09i·19-s + (−1.52 − 0.336i)20-s + (−0.708 + 0.569i)22-s + 1.05·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.444 + 0.895i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ 0.444 + 0.895i)$
$L(1)$  $\approx$  $0.9731237583$
$L(\frac12)$  $\approx$  $0.9731237583$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.885 + 1.10i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 3.50iT - 5T^{2} \)
11 \( 1 + 3.01iT - 11T^{2} \)
13 \( 1 + 3.90iT - 13T^{2} \)
17 \( 1 + 1.38T + 17T^{2} \)
19 \( 1 + 4.79iT - 19T^{2} \)
23 \( 1 - 5.06T + 23T^{2} \)
29 \( 1 - 4.91iT - 29T^{2} \)
31 \( 1 - 1.13T + 31T^{2} \)
37 \( 1 + 9.45iT - 37T^{2} \)
41 \( 1 + 4.11T + 41T^{2} \)
43 \( 1 - 1.51iT - 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 - 0.431iT - 53T^{2} \)
59 \( 1 - 7.40iT - 59T^{2} \)
61 \( 1 + 12.9iT - 61T^{2} \)
67 \( 1 + 3.36iT - 67T^{2} \)
71 \( 1 - 6.26T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 - 0.818T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−9.370732270710389582900341365334, −8.745063906441937106665681352107, −7.71093983998989599108611085824, −7.06929298819701133991808954577, −6.33501281039154305027355957847, −5.14268297378742439258909321443, −3.67745303046872908313176155489, −3.07817013692075480563272222326, −2.40474129518088803893733552345, −0.58652345978502163813270480630, 1.03723454490197360060397971843, 2.05060292219297249234001177414, 4.08855811836474116825220204766, 4.73784018754347985745717830156, 5.51671626424970724423545927203, 6.49268184826577712425353832257, 7.22317705608520466780163559340, 8.227785247397323431162523986677, 8.713310864124013277207274860236, 9.594103953312181968085226748206

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