Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Sign $0.887 - 0.461i$
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 + (−0.614 + 2.57i)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 + (−0.232 + 0.972i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.887 - 0.461i$
motivic weight  =  \(1\)
character  :  $\chi_{1680} (433, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1680,\ (\ :1/2),\ 0.887 - 0.461i)\)
\(L(1)\)  \(\approx\)  \(2.350041426\)
\(L(\frac12)\)  \(\approx\)  \(2.350041426\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.614 - 2.57i)T \)
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

−9.266895449303878172551440031315, −8.770516941004815860656211431341, −7.80369460536198924727885141570, −6.97649917522793822628385529120, −6.03251374619028176922211445988, −5.84175672121484702931292188624, −4.24266366447702310782601921755, −3.16139928007197552948113100464, −2.54507207899540097527357870546, −1.29168434142689295102366881641, 1.02205076281411068463562019929, 2.02541680437625225546900250374, 3.71288772739513424561379935963, 4.07641030639433167364953672435, 5.00166549665087971078513031043, 6.21001291352217136838431136526, 6.74437069683709161568425492373, 7.85559335627840692992457889399, 8.810274442378678286435846144080, 9.132492428835346627449042827129

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