Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Sign $0.443 + 0.896i$
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 + (1.28 − 1.82i)5-s + (1.75 − 1.97i)7-s − 1.00i·9-s + 2.67·11-s + (−1.22 + 1.22i)13-s + (−0.379 − 2.20i)15-s + (4.74 + 4.74i)17-s + 6.01·19-s + (−0.152 − 2.64i)21-s + (0.175 + 0.175i)23-s + (−1.67 − 4.71i)25-s + (−0.707 − 0.707i)27-s + 0.304i·29-s + 7.25i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.576 − 0.816i)5-s + (0.665 − 0.746i)7-s − 0.333i·9-s + 0.805·11-s + (−0.340 + 0.340i)13-s + (−0.0979 − 0.568i)15-s + (1.15 + 1.15i)17-s + 1.38·19-s + (−0.0332 − 0.576i)21-s + (0.0366 + 0.0366i)23-s + (−0.334 − 0.942i)25-s + (−0.136 − 0.136i)27-s + 0.0566i·29-s + 1.30i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.443 + 0.896i$
motivic weight  =  \(1\)
character  :  $\chi_{1680} (433, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1680,\ (\ :1/2),\ 0.443 + 0.896i)\)
\(L(1)\)  \(\approx\)  \(2.628468176\)
\(L(\frac12)\)  \(\approx\)  \(2.628468176\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.28 + 1.82i)T \)
7 \( 1 + (-1.75 + 1.97i)T \)
good11 \( 1 - 2.67T + 11T^{2} \)
13 \( 1 + (1.22 - 1.22i)T - 13iT^{2} \)
17 \( 1 + (-4.74 - 4.74i)T + 17iT^{2} \)
19 \( 1 - 6.01T + 19T^{2} \)
23 \( 1 + (-0.175 - 0.175i)T + 23iT^{2} \)
29 \( 1 - 0.304iT - 29T^{2} \)
31 \( 1 - 7.25iT - 31T^{2} \)
37 \( 1 + (0.735 - 0.735i)T - 37iT^{2} \)
41 \( 1 + 7.05iT - 41T^{2} \)
43 \( 1 + (0.304 + 0.304i)T + 43iT^{2} \)
47 \( 1 + (-0.556 - 0.556i)T + 47iT^{2} \)
53 \( 1 + (4.99 + 4.99i)T + 53iT^{2} \)
59 \( 1 + 7.98T + 59T^{2} \)
61 \( 1 - 5.53iT - 61T^{2} \)
67 \( 1 + (-3.43 + 3.43i)T - 67iT^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + (10.0 - 10.0i)T - 73iT^{2} \)
79 \( 1 + 11.2iT - 79T^{2} \)
83 \( 1 + (4.88 - 4.88i)T - 83iT^{2} \)
89 \( 1 + 6.91T + 89T^{2} \)
97 \( 1 + (-8.84 - 8.84i)T + 97iT^{2} \)
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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

−9.113652889425262566616338964686, −8.454437991625453790103455102425, −7.65921548989232211042052090270, −6.96968852762326821981473091779, −5.92814476267023460375895759934, −5.11942504661229295076668311226, −4.18154183618513578307252095891, −3.25302881929121552535557613418, −1.68426389345896051208933749877, −1.17258642491509972691804441691, 1.46351429508349018506931017905, 2.70734530473096481851232177469, 3.26633614749008050068935470052, 4.59281120671920182281720451328, 5.46312973656750310720058812404, 6.10381676034149076382538479630, 7.34900779560541395836954138467, 7.75580883748965618998945175195, 8.891515113565930553847051921680, 9.649664456675196608220089494129

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