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Properties

Label 2-1008-252.103-c1-0-41
Degree $2$
Conductor $1008$
Sign $0.708 + 0.705i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.64 + 0.551i)3-s + (2.87 − 1.65i)5-s + (−2.01 − 1.71i)7-s + (2.39 + 1.81i)9-s + (−4.94 − 2.85i)11-s + (−0.000820 − 0.000473i)13-s + (5.62 − 1.13i)15-s + (0.287 − 0.165i)17-s + (3.59 − 6.23i)19-s + (−2.35 − 3.92i)21-s + (5.37 − 3.10i)23-s + (2.99 − 5.18i)25-s + (2.92 + 4.29i)27-s + (5.06 + 8.77i)29-s + 3.88·31-s + ⋯
L(s)  = 1  + (0.947 + 0.318i)3-s + (1.28 − 0.741i)5-s + (−0.760 − 0.648i)7-s + (0.796 + 0.604i)9-s + (−1.49 − 0.860i)11-s + (−0.000227 − 0.000131i)13-s + (1.45 − 0.293i)15-s + (0.0697 − 0.0402i)17-s + (0.825 − 1.43i)19-s + (−0.514 − 0.857i)21-s + (1.12 − 0.647i)23-s + (0.599 − 1.03i)25-s + (0.562 + 0.826i)27-s + (0.940 + 1.62i)29-s + 0.696·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.708 + 0.705i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.708 + 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.446984018\)
\(L(\frac12)\) \(\approx\) \(2.446984018\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 0.551i)T \)
7 \( 1 + (2.01 + 1.71i)T \)
good5 \( 1 + (-2.87 + 1.65i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.94 + 2.85i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.000820 + 0.000473i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.287 + 0.165i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.59 + 6.23i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.37 + 3.10i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.06 - 8.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.88T + 31T^{2} \)
37 \( 1 + (-0.341 + 0.592i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.71 + 3.29i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.16 - 3.55i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.93T + 47T^{2} \)
53 \( 1 + (0.211 + 0.366i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.323T + 59T^{2} \)
61 \( 1 - 0.966iT - 61T^{2} \)
67 \( 1 - 8.48iT - 67T^{2} \)
71 \( 1 - 3.04iT - 71T^{2} \)
73 \( 1 + (0.293 - 0.169i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.5iT - 79T^{2} \)
83 \( 1 + (2.46 + 4.26i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.40 + 4.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-15.1 + 8.77i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.951331439219738879358024686437, −8.913170739607579240287807373424, −8.580703454567071484631676363579, −7.37172231323606816407103177703, −6.55349970038959808231729418953, −5.24371828726901184020882608368, −4.82034221896567337733475038963, −3.19022018470305404466787656528, −2.65630296298092212085937376676, −1.04784406521140949613346799274, 1.81513674802511418535920024780, 2.66114054350486125887307107590, 3.30558117962179744971550874564, 4.97812297076810968991717158746, 5.94237070248273632191335917222, 6.68205192661541603311122215009, 7.59670113065094897418483649610, 8.362712459129223643943855475767, 9.539687378394942176246468674781, 9.921639603925692403424718610866

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