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Properties

Label 2-1008-21.11-c2-0-12
Degree $2$
Conductor $1008$
Sign $0.735 + 0.678i$
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  + (−6.60 + 3.81i)5-s + (−4.89 + 5.00i)7-s + (−4.41 − 2.55i)11-s − 20.7·13-s + (−22.7 − 13.1i)17-s + (11.1 + 19.3i)19-s + (34.4 − 19.8i)23-s + (16.5 − 28.6i)25-s − 7.62i·29-s + (−5.89 + 10.2i)31-s + (13.2 − 51.7i)35-s + (30.1 + 52.2i)37-s − 11.8i·41-s + 30.3·43-s + (−33.0 + 19.0i)47-s + ⋯
L(s)  = 1  + (−1.32 + 0.762i)5-s + (−0.698 + 0.715i)7-s + (−0.401 − 0.231i)11-s − 1.59·13-s + (−1.33 − 0.772i)17-s + (0.588 + 1.01i)19-s + (1.49 − 0.864i)23-s + (0.662 − 1.14i)25-s − 0.262i·29-s + (−0.190 + 0.329i)31-s + (0.377 − 1.47i)35-s + (0.815 + 1.41i)37-s − 0.287i·41-s + 0.705·43-s + (−0.703 + 0.406i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.678i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.735 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.735 + 0.678i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.735 + 0.678i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5031417171\)
\(L(\frac12)\) \(\approx\) \(0.5031417171\)
\(L(2)\) 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 \( 1 + (4.89 - 5.00i)T \)
good5 \( 1 + (6.60 - 3.81i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (4.41 + 2.55i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 20.7T + 169T^{2} \)
17 \( 1 + (22.7 + 13.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.1 - 19.3i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-34.4 + 19.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 7.62iT - 841T^{2} \)
31 \( 1 + (5.89 - 10.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-30.1 - 52.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 11.8iT - 1.68e3T^{2} \)
43 \( 1 - 30.3T + 1.84e3T^{2} \)
47 \( 1 + (33.0 - 19.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-5.11 - 2.95i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-38.2 - 22.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-24.7 - 42.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-31.5 + 54.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 41.6iT - 5.04e3T^{2} \)
73 \( 1 + (-2.08 + 3.61i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (20.8 + 36.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 145. iT - 6.88e3T^{2} \)
89 \( 1 + (-38.0 + 21.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 39.5T + 9.40e3T^{2} \)
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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

−9.681926209621724600665523553701, −8.831405498427282449378651827062, −7.88716117093450793168406196043, −7.14245949000903739042369054388, −6.52542315200718014136794662699, −5.22510636745095624598285391206, −4.35471012723059153899176219491, −3.07381456516557522283474125998, −2.59198643591875840321971640276, −0.24977177522808240900656016071, 0.69899801794098150488428919127, 2.55499915935785912976472903486, 3.74727824913826700989531552858, 4.55047023002565956758844254188, 5.26410401723004467612459383361, 6.87683666442733337488876699517, 7.29193093111925099019513744434, 8.093614803885897424098049912232, 9.150142319521004669679994371762, 9.657186340551797772979889702735

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