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Properties

Label 2-1008-28.11-c2-0-28
Degree $2$
Conductor $1008$
Sign $0.567 + 0.823i$
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  + (5.5 − 4.33i)7-s + 13-s + (7.5 − 4.33i)19-s + (12.5 − 21.6i)25-s + (16.5 + 9.52i)31-s + (−23.5 − 40.7i)37-s + 22.5i·43-s + (11.4 − 47.6i)49-s + (−37 − 64.0i)61-s + (67.5 + 38.9i)67-s + (71.5 − 123. i)73-s + (76.5 − 44.1i)79-s + (5.5 − 4.33i)91-s − 2·97-s + (175.5 − 101. i)103-s + ⋯
L(s)  = 1  + (0.785 − 0.618i)7-s + 0.0769·13-s + (0.394 − 0.227i)19-s + (0.5 − 0.866i)25-s + (0.532 + 0.307i)31-s + (−0.635 − 1.10i)37-s + 0.523i·43-s + (0.234 − 0.972i)49-s + (−0.606 − 1.05i)61-s + (1.00 + 0.581i)67-s + (0.979 − 1.69i)73-s + (0.968 − 0.559i)79-s + (0.0604 − 0.0475i)91-s − 0.0206·97-s + (1.70 − 0.983i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.567 + 0.823i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.567 + 0.823i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.010077610\)
\(L(\frac12)\) \(\approx\) \(2.010077610\)
\(L(2)\) 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 \( 1 + (-5.5 + 4.33i)T \)
good5 \( 1 + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (60.5 + 104. i)T^{2} \)
13 \( 1 - T + 169T^{2} \)
17 \( 1 + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-7.5 + 4.33i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (264.5 - 458. i)T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + (-16.5 - 9.52i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (23.5 + 40.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 1.68e3T^{2} \)
43 \( 1 - 22.5iT - 1.84e3T^{2} \)
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 \( 1 + (-67.5 - 38.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + (-71.5 + 123. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-76.5 + 44.1i)T + (3.12e3 - 5.40e3i)T^{2} \)
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

−9.688674582938974965289471917340, −8.732135179380581466644811093375, −7.972336640238062003791035080807, −7.19220413681395708224900008111, −6.29877878675697719137061976468, −5.15737543731311386607988806114, −4.42230554919425283514563195854, −3.33280901279128916002537780361, −2.00501447327686400116580184277, −0.71400108496781758905258083498, 1.22438235283380467097244386369, 2.41846362803239515780470932483, 3.58569337890748818679099753870, 4.79917641359029750314164115540, 5.47372684064139686807866684122, 6.48682087924339835809378581906, 7.48153563487650918239760894922, 8.288084464190983669019018713015, 8.987880373481541430910861386099, 9.865567510382839076045297837573

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