Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.832 + 0.553i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.35 − 1.93i)5-s + (2.5 − 0.866i)7-s + (3.35 + 1.93i)11-s − 3.46i·13-s + (2 + 3.46i)19-s + (−6.70 + 3.87i)23-s + (5.00 − 8.66i)25-s − 6.70·29-s + (−0.5 + 0.866i)31-s + (6.70 − 7.74i)35-s + (−2 − 3.46i)37-s + 7.74i·41-s + 6.92i·43-s + (−6.70 − 11.6i)47-s + (5.5 − 4.33i)49-s + ⋯
L(s)  = 1  + (1.50 − 0.866i)5-s + (0.944 − 0.327i)7-s + (1.01 + 0.583i)11-s − 0.960i·13-s + (0.458 + 0.794i)19-s + (−1.39 + 0.807i)23-s + (1.00 − 1.73i)25-s − 1.24·29-s + (−0.0898 + 0.155i)31-s + (1.13 − 1.30i)35-s + (−0.328 − 0.569i)37-s + 1.20i·41-s + 1.05i·43-s + (−0.978 − 1.69i)47-s + (0.785 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.832 + 0.553i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (703, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.832 + 0.553i)\)
\(L(1)\)  \(\approx\)  \(2.385255340\)
\(L(\frac12)\)  \(\approx\)  \(2.385255340\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (-3.35 + 1.93i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.35 - 1.93i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.70 - 3.87i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.70T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.74iT - 41T^{2} \)
43 \( 1 - 6.92iT - 43T^{2} \)
47 \( 1 + (6.70 + 11.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.35 - 5.80i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.35 - 5.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9 + 5.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 3.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.74iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.5 + 6.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 + (-6.70 + 3.87i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−9.799266901010023586090539598515, −9.241266803621624269415961861260, −8.244479611085667202216951880944, −7.52248382687598867074062565644, −6.24160943624386512154633614810, −5.55454660153073689525482995804, −4.80952471233036566344778781263, −3.70426249213357988878858868000, −1.96455769706301333888722050995, −1.34475117445146891390206886479, 1.64274280156998604881553889558, 2.36129205700071365374079727117, 3.72506842785513394010179512699, 4.94654813181701953510236998979, 5.92297430354469545985470702092, 6.48814240683922879799795106919, 7.37619543517435020569116353335, 8.614445382978127797714007138123, 9.261941854307159834179319249370, 9.952906981415931709743554420855

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