Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Sign $0.957 - 0.288i$
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 + (2.23 − 0.0836i)5-s + (2.64 − 0.0627i)7-s − 1.00i·9-s − 3.98·11-s + (0.500 − 0.500i)13-s + (−1.52 + 1.63i)15-s + (−1.67 − 1.67i)17-s + 7.21·19-s + (−1.82 + 1.91i)21-s + (5.16 + 5.16i)23-s + (4.98 − 0.373i)25-s + (0.707 + 0.707i)27-s + 3.65i·29-s − 4.93i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.999 − 0.0373i)5-s + (0.999 − 0.0237i)7-s − 0.333i·9-s − 1.20·11-s + (0.138 − 0.138i)13-s + (−0.392 + 0.423i)15-s + (−0.407 − 0.407i)17-s + 1.65·19-s + (−0.398 + 0.417i)21-s + (1.07 + 1.07i)23-s + (0.997 − 0.0747i)25-s + (0.136 + 0.136i)27-s + 0.678i·29-s − 0.886i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.957 - 0.288i$
motivic weight  =  \(1\)
character  :  $\chi_{1680} (433, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1680,\ (\ :1/2),\ 0.957 - 0.288i)\)
\(L(1)\)  \(\approx\)  \(2.005890217\)
\(L(\frac12)\)  \(\approx\)  \(2.005890217\)
\(L(\frac{3}{2})\)   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 \( 1 + (-2.23 + 0.0836i)T \)
7 \( 1 + (-2.64 + 0.0627i)T \)
good11 \( 1 + 3.98T + 11T^{2} \)
13 \( 1 + (-0.500 + 0.500i)T - 13iT^{2} \)
17 \( 1 + (1.67 + 1.67i)T + 17iT^{2} \)
19 \( 1 - 7.21T + 19T^{2} \)
23 \( 1 + (-5.16 - 5.16i)T + 23iT^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 + 4.93iT - 31T^{2} \)
37 \( 1 + (-0.292 + 0.292i)T - 37iT^{2} \)
41 \( 1 + 7.63iT - 41T^{2} \)
43 \( 1 + (3.65 + 3.65i)T + 43iT^{2} \)
47 \( 1 + (-0.305 - 0.305i)T + 47iT^{2} \)
53 \( 1 + (-5.39 - 5.39i)T + 53iT^{2} \)
59 \( 1 + 6.10T + 59T^{2} \)
61 \( 1 - 7.11iT - 61T^{2} \)
67 \( 1 + (0.944 - 0.944i)T - 67iT^{2} \)
71 \( 1 + 1.19T + 71T^{2} \)
73 \( 1 + (-1.38 + 1.38i)T - 73iT^{2} \)
79 \( 1 - 8.64iT - 79T^{2} \)
83 \( 1 + (-11.9 + 11.9i)T - 83iT^{2} \)
89 \( 1 - 7.82T + 89T^{2} \)
97 \( 1 + (-7.43 - 7.43i)T + 97iT^{2} \)
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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

−9.374538021694822477603182709158, −8.825270147664462902440347446904, −7.66566670851780137733963543360, −7.13290865342012085964367680693, −5.79759572386708552602248032767, −5.29522642133135004488439989804, −4.78592384315084018443960756523, −3.36148955979407993695996775412, −2.31346891834334979940297797602, −1.07539600389118560525244872380, 1.06581015204560507918505721862, 2.11787839723481740388908280199, 3.05823121146043875506090694981, 4.80828652148871820137763049238, 5.11624300252636047356916427095, 6.04587906206591642184191227980, 6.86970595524963592843753446309, 7.75171624118850950317835968251, 8.424562072346381531121610857259, 9.329879070251381182000137235499

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