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Properties

Label 2-1008-7.5-c2-0-22
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: $\chi_{1008} (145, \cdot )$
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

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

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