Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Sign $0.812 + 0.583i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)3-s + (−1.03 − 1.98i)5-s + (−2.57 + 0.614i)7-s − 1.00i·9-s + 3.85·11-s + (−3.66 + 3.66i)13-s + (2.13 + 0.668i)15-s + (1.49 + 1.49i)17-s − 0.0697·19-s + (1.38 − 2.25i)21-s + (0.534 + 0.534i)23-s + (−2.85 + 4.10i)25-s + (0.707 + 0.707i)27-s − 2.77i·29-s − 2.39i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.463 − 0.886i)5-s + (−0.972 + 0.232i)7-s − 0.333i·9-s + 1.16·11-s + (−1.01 + 1.01i)13-s + (0.550 + 0.172i)15-s + (0.361 + 0.361i)17-s − 0.0160·19-s + (0.302 − 0.491i)21-s + (0.111 + 0.111i)23-s + (−0.570 + 0.821i)25-s + (0.136 + 0.136i)27-s − 0.514i·29-s − 0.430i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.812 + 0.583i$
motivic weight  =  \(1\)
character  :  $\chi_{1680} (433, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1680,\ (\ :1/2),\ 0.812 + 0.583i)\)
\(L(1)\)  \(\approx\)  \(1.011275805\)
\(L(\frac12)\)  \(\approx\)  \(1.011275805\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.03 + 1.98i)T \)
7 \( 1 + (2.57 - 0.614i)T \)
good11 \( 1 - 3.85T + 11T^{2} \)
13 \( 1 + (3.66 - 3.66i)T - 13iT^{2} \)
17 \( 1 + (-1.49 - 1.49i)T + 17iT^{2} \)
19 \( 1 + 0.0697T + 19T^{2} \)
23 \( 1 + (-0.534 - 0.534i)T + 23iT^{2} \)
29 \( 1 + 2.77iT - 29T^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 + (-6.18 + 6.18i)T - 37iT^{2} \)
41 \( 1 + 8.68iT - 41T^{2} \)
43 \( 1 + (-2.77 - 2.77i)T + 43iT^{2} \)
47 \( 1 + (5.49 + 5.49i)T + 47iT^{2} \)
53 \( 1 + (-6.13 - 6.13i)T + 53iT^{2} \)
59 \( 1 - 6.97T + 59T^{2} \)
61 \( 1 + 14.3iT - 61T^{2} \)
67 \( 1 + (0.416 - 0.416i)T - 67iT^{2} \)
71 \( 1 - 8.12T + 71T^{2} \)
73 \( 1 + (-9.55 + 9.55i)T - 73iT^{2} \)
79 \( 1 + 9.86iT - 79T^{2} \)
83 \( 1 + (1.63 - 1.63i)T - 83iT^{2} \)
89 \( 1 + 5.05T + 89T^{2} \)
97 \( 1 + (-6.85 - 6.85i)T + 97iT^{2} \)
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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

−9.465251079214858623309470965730, −8.740898683387521618506820359303, −7.64475890371555269198761699718, −6.78428116076732834517532960397, −6.02780044166218980423601329179, −5.11979560353563701026058248696, −4.16883549462545929230025564552, −3.64981016566351733031749790162, −2.10667435463428702124867969665, −0.56144630878097495566190186065, 0.884317354470044676764357256191, 2.62678110352628215550065515567, 3.33205575861112184024982206769, 4.36565931053256229872404282370, 5.52322629407407085970496062667, 6.44890311128371903340927844494, 6.94586168108628043209297936331, 7.61214381155761479876355324416, 8.526807137417916875286622156523, 9.819197945497410059979715145625

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