Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.667 - 0.744i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 + 0.866i)7-s + 5.19i·13-s + (0.5 + 0.866i)19-s + (−2.5 + 4.33i)25-s + (5.5 − 9.52i)31-s + (5.5 + 9.52i)37-s + 1.73i·43-s + (5.5 + 4.33i)49-s + (−6 + 3.46i)61-s + (13.5 + 7.79i)67-s + (1.5 + 0.866i)73-s + (4.5 − 2.59i)79-s + (−4.5 + 12.9i)91-s + 13.8i·97-s + (−6.5 − 11.2i)103-s + ⋯
L(s)  = 1  + (0.944 + 0.327i)7-s + 1.44i·13-s + (0.114 + 0.198i)19-s + (−0.5 + 0.866i)25-s + (0.987 − 1.71i)31-s + (0.904 + 1.56i)37-s + 0.264i·43-s + (0.785 + 0.618i)49-s + (−0.768 + 0.443i)61-s + (1.64 + 0.952i)67-s + (0.175 + 0.101i)73-s + (0.506 − 0.292i)79-s + (−0.471 + 1.36i)91-s + 1.40i·97-s + (−0.640 − 1.10i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.667 - 0.744i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (703, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.667 - 0.744i)\)
\(L(1)\)  \(\approx\)  \(1.713671318\)
\(L(\frac12)\)  \(\approx\)  \(1.713671318\)
\(L(\frac{3}{2})\)   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 \( 1 + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-5.5 + 9.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.5 - 7.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−9.907146378929162580932039969213, −9.307985346581686852034792200507, −8.353888459261358224880990529514, −7.69646155047672918152114601450, −6.67589517465499948367453876473, −5.79249863017645001869900063218, −4.74671477932337098703228579638, −4.01589421443877902256444259515, −2.53215073632567230730026855483, −1.45580002141998516231787664156, 0.872390336948986254722597598955, 2.34767897603019312967976076449, 3.54829624376638419107881016540, 4.67850642741028613609107877559, 5.42368296416434588925194671934, 6.44295782931424713837663354124, 7.56052952340716718020469402504, 8.084055615272246010021346641484, 8.908570292508444541811797263013, 10.04369075740283898964419154714

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