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Properties

Label 2-1008-28.23-c2-0-13
Degree $2$
Conductor $1008$
Sign $0.978 - 0.205i$
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  + (−6.5 − 2.59i)7-s − 23·13-s + (31.5 + 18.1i)19-s + (12.5 + 21.6i)25-s + (52.5 − 30.3i)31-s + (36.5 − 63.2i)37-s + 60.6i·43-s + (35.5 + 33.7i)49-s + (−37 + 64.0i)61-s + (115.5 − 66.6i)67-s + (−48.5 − 84.0i)73-s + (136.5 + 78.8i)79-s + (149.5 + 59.7i)91-s − 2·97-s + (115.5 + 66.6i)103-s + ⋯
L(s)  = 1  + (−0.928 − 0.371i)7-s − 1.76·13-s + (1.65 + 0.957i)19-s + (0.5 + 0.866i)25-s + (1.69 − 0.977i)31-s + (0.986 − 1.70i)37-s + 1.40i·43-s + (0.724 + 0.689i)49-s + (−0.606 + 1.05i)61-s + (1.72 − 0.995i)67-s + (−0.664 − 1.15i)73-s + (1.72 + 0.997i)79-s + (1.64 + 0.656i)91-s − 0.0206·97-s + (1.12 + 0.647i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.978 - 0.205i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.978 - 0.205i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.457764130\)
\(L(\frac12)\) \(\approx\) \(1.457764130\)
\(L(2)\) 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 \( 1 + (6.5 + 2.59i)T \)
good5 \( 1 + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 + 23T + 169T^{2} \)
17 \( 1 + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-31.5 - 18.1i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (264.5 + 458. i)T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + (-52.5 + 30.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-36.5 + 63.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 1.68e3T^{2} \)
43 \( 1 - 60.6iT - 1.84e3T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (37 - 64.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-115.5 + 66.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + (48.5 + 84.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-136.5 - 78.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 2T + 9.40e3T^{2} \)
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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

−9.607931730465641229219615915647, −9.407902740988887131470410474826, −7.82511347775379273322023904627, −7.43859717966742683474164767387, −6.42855365514036141817447092960, −5.49667760997326128824157965397, −4.52493703889570213653621093727, −3.39364825481396729155089702219, −2.48400863988572292614443983373, −0.801376785678600624648744206212, 0.67348264233739362812604293282, 2.52829865476815706430762051205, 3.13370019881569468723694529413, 4.64058970352081360829039636190, 5.28879450653481454303764304204, 6.51892863584218780991023278632, 7.08160449567698371052630204561, 8.068127096304707317908282848556, 9.063933410034928767848994603131, 9.840553007151080042388679141353

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