Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7 $
Sign $-0.220 + 0.975i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.41i)3-s + (0.414 + 0.414i)5-s + 7-s + (−1.00 − 2.82i)9-s + (3.82 − 3.82i)11-s + (−4.41 − 4.41i)13-s + (1 − 0.171i)15-s + 1.17i·17-s + (−0.414 + 0.414i)19-s + (1 − 1.41i)21-s + 7.65i·23-s − 4.65i·25-s + (−5.00 − 1.41i)27-s + (−3 + 3i)29-s − 6.48i·31-s + ⋯
L(s)  = 1  + (0.577 − 0.816i)3-s + (0.185 + 0.185i)5-s + 0.377·7-s + (−0.333 − 0.942i)9-s + (1.15 − 1.15i)11-s + (−1.22 − 1.22i)13-s + (0.258 − 0.0442i)15-s + 0.284i·17-s + (−0.0950 + 0.0950i)19-s + (0.218 − 0.308i)21-s + 1.59i·23-s − 0.931i·25-s + (−0.962 − 0.272i)27-s + (−0.557 + 0.557i)29-s − 1.16i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $-0.220 + 0.975i$
motivic weight  =  \(1\)
character  :  $\chi_{1344} (239, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1344,\ (\ :1/2),\ -0.220 + 0.975i)$
$L(1)$  $\approx$  $2.016198138$
$L(\frac12)$  $\approx$  $2.016198138$
$L(\frac{3}{2})$   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 \( 1 \)
3 \( 1 + (-1 + 1.41i)T \)
7 \( 1 - T \)
good5 \( 1 + (-0.414 - 0.414i)T + 5iT^{2} \)
11 \( 1 + (-3.82 + 3.82i)T - 11iT^{2} \)
13 \( 1 + (4.41 + 4.41i)T + 13iT^{2} \)
17 \( 1 - 1.17iT - 17T^{2} \)
19 \( 1 + (0.414 - 0.414i)T - 19iT^{2} \)
23 \( 1 - 7.65iT - 23T^{2} \)
29 \( 1 + (3 - 3i)T - 29iT^{2} \)
31 \( 1 + 6.48iT - 31T^{2} \)
37 \( 1 + (-1.82 + 1.82i)T - 37iT^{2} \)
41 \( 1 + 0.343T + 41T^{2} \)
43 \( 1 + (-3.82 - 3.82i)T + 43iT^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 + (5.82 + 5.82i)T + 53iT^{2} \)
59 \( 1 + (-10.0 + 10.0i)T - 59iT^{2} \)
61 \( 1 + (-2.41 - 2.41i)T + 61iT^{2} \)
67 \( 1 + (-7 + 7i)T - 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 - 5.17iT - 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + (8.89 + 8.89i)T + 83iT^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.344858800182026657760251821710, −8.410027824286011466515270918645, −7.81846925467028719906708392621, −7.05959415972599663549161393469, −6.08437173250393318372872227371, −5.44777015189044405786635025240, −3.95588747053071547790312406590, −3.10587736286391075381991735721, −2.05195571499784628574937193673, −0.77653286278372803848867569460, 1.76359661864946125204266022517, 2.63467017174363341050064987706, 4.16111010945247825257619121064, 4.47637591802946149056106556558, 5.40913998672384553376734622346, 6.80822121244550580333894949840, 7.31587205335923583245959363050, 8.486220670472092551301281997938, 9.211802960277382876001919574040, 9.621694745724810446415068924141

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