Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Sign $-0.701 + 0.712i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (−2 − 1.73i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 13-s − 0.999·15-s + (−1.5 − 2.59i)19-s + (2.5 − 0.866i)21-s + (3.5 + 6.06i)23-s + (−0.499 + 0.866i)25-s + 0.999·27-s − 8·29-s + (−1 + 1.73i)31-s + (−0.499 − 0.866i)33-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (−0.755 − 0.654i)7-s + (−0.166 − 0.288i)9-s + (−0.150 + 0.261i)11-s + 0.277·13-s − 0.258·15-s + (−0.344 − 0.596i)19-s + (0.545 − 0.188i)21-s + (0.729 + 1.26i)23-s + (−0.0999 + 0.173i)25-s + 0.192·27-s − 1.48·29-s + (−0.179 + 0.311i)31-s + (−0.0870 − 0.150i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.701 + 0.712i$
motivic weight  =  \(1\)
character  :  $\chi_{1680} (1201, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1680,\ (\ :1/2),\ -0.701 + 0.712i)\)
\(L(1)\)  \(\approx\)  \(0.2200321770\)
\(L(\frac12)\)  \(\approx\)  \(0.2200321770\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2 + 1.73i)T \)
good11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (2.5 + 4.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.5 + 9.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16T + 97T^{2} \)
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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

−9.309689410272044474232605311289, −8.363259098663911736200420303808, −7.08918743127239478044254391619, −6.89001979491523585099234649437, −5.70158173067212268174121192934, −5.04168140523126230106896089692, −3.78222559471431756922802333872, −3.30921537606468101755556365988, −1.84697273236623692140284270232, −0.084000489316477839434140689195, 1.49137808468798994103135524713, 2.63533721770063924908287739903, 3.64969016491551752799892052416, 4.92545889399787184421972231256, 5.67792637778453247867409500125, 6.41807598244263102116197279392, 7.07423613665469414586408719283, 8.312083123535375178383890198474, 8.666328106996505670956017161449, 9.645529194206775674237179521887

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