Properties

Label 2-1008-7.5-c2-0-33
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: $\chi_{1008} (145, \cdot )$
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

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

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