Properties

Label 2-1008-7.3-c2-0-8
Degree $2$
Conductor $1008$
Sign $-0.989 - 0.144i$
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  + (−7.24 + 4.18i)5-s + (6.74 + 1.88i)7-s + (−3 + 5.19i)11-s + 17.8i·13-s + (16.2 + 9.37i)17-s + (14.7 − 8.51i)19-s + (−6.72 − 11.6i)23-s + (22.4 − 38.9i)25-s − 33.9·29-s + (−12.7 − 7.37i)31-s + (−56.6 + 14.5i)35-s + (−2.98 − 5.17i)37-s + 35.2i·41-s − 15.4·43-s + (−28.7 + 16.6i)47-s + ⋯
L(s)  = 1  + (−1.44 + 0.836i)5-s + (0.963 + 0.268i)7-s + (−0.272 + 0.472i)11-s + 1.37i·13-s + (0.955 + 0.551i)17-s + (0.775 − 0.447i)19-s + (−0.292 − 0.506i)23-s + (0.898 − 1.55i)25-s − 1.17·29-s + (−0.412 − 0.237i)31-s + (−1.61 + 0.416i)35-s + (−0.0806 − 0.139i)37-s + 0.859i·41-s − 0.360·43-s + (−0.611 + 0.353i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.989 - 0.144i$
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: \((2,\ 1008,\ (\ :1),\ -0.989 - 0.144i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8022584543\)
\(L(\frac12)\) \(\approx\) \(0.8022584543\)
\(L(2)\) 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 \( 1 + (-6.74 - 1.88i)T \)
good5 \( 1 + (7.24 - 4.18i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 17.8iT - 169T^{2} \)
17 \( 1 + (-16.2 - 9.37i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.7 + 8.51i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (6.72 + 11.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 33.9T + 841T^{2} \)
31 \( 1 + (12.7 + 7.37i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (2.98 + 5.17i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 35.2iT - 1.68e3T^{2} \)
43 \( 1 + 15.4T + 1.84e3T^{2} \)
47 \( 1 + (28.7 - 16.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (17.2 - 29.9i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-23.6 - 13.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (34.9 - 20.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (57.1 - 99.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 18.6T + 5.04e3T^{2} \)
73 \( 1 + (101. + 58.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (44.1 + 76.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 75.7iT - 6.88e3T^{2} \)
89 \( 1 + (-18 + 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 30.5iT - 9.40e3T^{2} \)
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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

−10.32728710525188209193339168122, −9.265605148211006588962773677682, −8.327141681115605026052014871488, −7.55103162455783713141125607213, −7.14353048933846558797273590033, −5.93388825130317141385494498370, −4.70220822036929088952846573615, −4.02857144301866381675283536811, −2.95920362816107700621719535298, −1.63899884315848400907841425132, 0.27462356762991584577016125107, 1.32825653985260738028535179054, 3.22897611241325566439599794010, 3.92214693263181214410570996600, 5.16574387279536563007796201393, 5.46956820473119886712766667810, 7.27110696772688942599533259664, 7.914030786384420176657801593774, 8.173070325289605722014247042612, 9.246457929019942605619823320881

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