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Properties

Label 2-1008-21.2-c2-0-24
Degree $2$
Conductor $1008$
Sign $0.851 + 0.524i$
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  + (3.67 + 2.12i)5-s + (3.5 − 6.06i)7-s + (18.3 − 10.6i)11-s + 23·13-s + (−14.6 + 8.48i)17-s + (−0.5 + 0.866i)19-s + (−14.6 − 8.48i)23-s + (−3.5 − 6.06i)25-s + 33.9i·29-s + (−24.5 − 42.4i)31-s + (25.7 − 14.8i)35-s + (−8.5 + 14.7i)37-s − 21.2i·41-s − 47·43-s + (33.0 + 19.0i)47-s + ⋯
L(s)  = 1  + (0.734 + 0.424i)5-s + (0.5 − 0.866i)7-s + (1.67 − 0.964i)11-s + 1.76·13-s + (−0.864 + 0.499i)17-s + (−0.0263 + 0.0455i)19-s + (−0.638 − 0.368i)23-s + (−0.140 − 0.242i)25-s + 1.17i·29-s + (−0.790 − 1.36i)31-s + (0.734 − 0.424i)35-s + (−0.229 + 0.397i)37-s − 0.517i·41-s − 1.09·43-s + (0.703 + 0.406i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.851 + 0.524i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.851 + 0.524i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.708154471\)
\(L(\frac12)\) \(\approx\) \(2.708154471\)
\(L(2)\) 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 \( 1 + (-3.5 + 6.06i)T \)
good5 \( 1 + (-3.67 - 2.12i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-18.3 + 10.6i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 23T + 169T^{2} \)
17 \( 1 + (14.6 - 8.48i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (14.6 + 8.48i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 33.9iT - 841T^{2} \)
31 \( 1 + (24.5 + 42.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (8.5 - 14.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 21.2iT - 1.68e3T^{2} \)
43 \( 1 + 47T + 1.84e3T^{2} \)
47 \( 1 + (-33.0 - 19.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-73.4 + 42.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-44.0 + 25.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-20 + 34.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-11.5 - 19.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 63.6iT - 5.04e3T^{2} \)
73 \( 1 + (8.5 + 14.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (39.5 - 68.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 106. iT - 6.88e3T^{2} \)
89 \( 1 + (-117. - 67.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 40T + 9.40e3T^{2} \)
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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

−9.724632926057579204900651463234, −8.731941815034314384733398640678, −8.303896694742834891327274285341, −6.85886707040224696681206280344, −6.39887313091552205671213153166, −5.60585303720514930301112748131, −4.02585206918778337785281556502, −3.68455445482901524178133188513, −1.95407775186069141000352699009, −0.982379228572056584076253720844, 1.37839573157699401027098016248, 2.04549936382176454066557562292, 3.67139399976482296891075856919, 4.57904563935108502431845680779, 5.66569265907299854544553028429, 6.29213762819488196518202065622, 7.21533437358769393317569388313, 8.565968226466092738811476000525, 8.961841821386361007771749379760, 9.597204622678598379078417514476

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