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Properties

Label 2-1008-7.3-c2-0-19
Degree $2$
Conductor $1008$
Sign $0.829 - 0.558i$
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  + (0.425 − 0.245i)5-s + (6.25 − 3.14i)7-s + (−5.17 + 8.97i)11-s + 8.96i·13-s + (−0.259 − 0.149i)17-s + (24.4 − 14.0i)19-s + (2.82 + 4.89i)23-s + (−12.3 + 21.4i)25-s − 25.4·29-s + (39.5 + 22.8i)31-s + (1.88 − 2.87i)35-s + (5.17 + 8.95i)37-s + 47.6i·41-s − 6.83·43-s + (56.4 − 32.5i)47-s + ⋯
L(s)  = 1  + (0.0850 − 0.0490i)5-s + (0.893 − 0.449i)7-s + (−0.470 + 0.815i)11-s + 0.689i·13-s + (−0.0152 − 0.00880i)17-s + (1.28 − 0.741i)19-s + (0.122 + 0.212i)23-s + (−0.495 + 0.857i)25-s − 0.877·29-s + (1.27 + 0.737i)31-s + (0.0539 − 0.0820i)35-s + (0.139 + 0.242i)37-s + 1.16i·41-s − 0.159·43-s + (1.20 − 0.693i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.829 - 0.558i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.829 - 0.558i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.085052229\)
\(L(\frac12)\) \(\approx\) \(2.085052229\)
\(L(2)\) 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 \( 1 + (-6.25 + 3.14i)T \)
good5 \( 1 + (-0.425 + 0.245i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (5.17 - 8.97i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 8.96iT - 169T^{2} \)
17 \( 1 + (0.259 + 0.149i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-24.4 + 14.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-2.82 - 4.89i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 25.4T + 841T^{2} \)
31 \( 1 + (-39.5 - 22.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-5.17 - 8.95i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 47.6iT - 1.68e3T^{2} \)
43 \( 1 + 6.83T + 1.84e3T^{2} \)
47 \( 1 + (-56.4 + 32.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-8.95 + 15.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (42.0 + 24.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (30 - 17.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-40.5 + 70.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 133.T + 5.04e3T^{2} \)
73 \( 1 + (5.51 + 3.18i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-26.5 - 45.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 116. iT - 6.88e3T^{2} \)
89 \( 1 + (-27.1 + 15.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 54.6iT - 9.40e3T^{2} \)
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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

−9.763100962079980956711453298589, −9.160117185587559512676752603414, −8.014072632077882041654980949614, −7.42452536185461768013943764760, −6.61714167015696860037238699229, −5.25533837617878611518620921571, −4.75947237698096092722457084148, −3.61919289304191080048422491607, −2.26828706536809326883316015728, −1.14927565953316961344909214641, 0.77032233226741555356924797098, 2.20429798019760435005774602718, 3.25500394593372632605595001396, 4.45876292933970677706680454411, 5.56423604913525782371154112052, 5.94624450794340216328193459746, 7.43723832769780698212613195496, 8.011325190315397714714190526184, 8.737706067761055245506572809033, 9.739148840314862709904640656026

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