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Properties

Label 2-1008-21.2-c2-0-14
Degree $2$
Conductor $1008$
Sign $0.389 - 0.921i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.33 + 3.08i)5-s + (−0.220 + 6.99i)7-s + (−4.23 + 2.44i)11-s + 15.1·13-s + (16.4 − 9.48i)17-s + (7.56 − 13.1i)19-s + (6.19 + 3.57i)23-s + (6.49 + 11.2i)25-s + 24.0i·29-s + (5.55 + 9.63i)31-s + (−22.7 + 36.6i)35-s + (−21.1 + 36.5i)37-s − 1.60i·41-s − 30.0·43-s + (58.5 + 33.8i)47-s + ⋯
L(s)  = 1  + (1.06 + 0.616i)5-s + (−0.0315 + 0.999i)7-s + (−0.385 + 0.222i)11-s + 1.16·13-s + (0.966 − 0.558i)17-s + (0.398 − 0.689i)19-s + (0.269 + 0.155i)23-s + (0.259 + 0.449i)25-s + 0.827i·29-s + (0.179 + 0.310i)31-s + (−0.649 + 1.04i)35-s + (−0.570 + 0.988i)37-s − 0.0391i·41-s − 0.698·43-s + (1.24 + 0.719i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.389 - 0.921i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.389 - 0.921i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.461580800\)
\(L(\frac12)\) \(\approx\) \(2.461580800\)
\(L(2)\) 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 \( 1 + (0.220 - 6.99i)T \)
good5 \( 1 + (-5.33 - 3.08i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (4.23 - 2.44i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 15.1T + 169T^{2} \)
17 \( 1 + (-16.4 + 9.48i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-7.56 + 13.1i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-6.19 - 3.57i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 24.0iT - 841T^{2} \)
31 \( 1 + (-5.55 - 9.63i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (21.1 - 36.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 1.60iT - 1.68e3T^{2} \)
43 \( 1 + 30.0T + 1.84e3T^{2} \)
47 \( 1 + (-58.5 - 33.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (3.83 - 2.21i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-74.5 + 43.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (38.1 - 66.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.50 + 4.34i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 24.9iT - 5.04e3T^{2} \)
73 \( 1 + (66.6 + 115. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (73.9 - 128. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 20.9iT - 6.88e3T^{2} \)
89 \( 1 + (-40.7 - 23.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 44.5T + 9.40e3T^{2} \)
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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

−9.918795789151717696317944189601, −9.149225548000164252014775991364, −8.438275728136066810413305659020, −7.29336147537530908782431322779, −6.40797817729873758728289313179, −5.66928401039278130527809329320, −4.97444533789524545262029162086, −3.31472532083474405327325388972, −2.58322164578330348756497598405, −1.37655799542518994388099327682, 0.852105633516750168768687250401, 1.79494170457485056093052305229, 3.35272060281995949259037959961, 4.24869568450951603834390292914, 5.53524800937507897147532713324, 5.93998432676459103189455316513, 7.09665294552064635257488108944, 8.049021593330409486056325262869, 8.769173566538325768010244925353, 9.791269408330713398084411584110

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