Properties

Degree $2$
Conductor $1008$
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

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.667 + 0.744i$
Motivic weight: \(1\)
Character: $\chi_{1008} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.667 + 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.713671318\)
\(L(\frac12)\) \(\approx\) \(1.713671318\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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