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Properties

Label 2-1008-28.3-c1-0-19
Degree $2$
Conductor $1008$
Sign $-0.991 + 0.126i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 0.866i)5-s − 2.64·7-s + (−3.96 − 2.29i)11-s + 3.46i·13-s + (−4.5 − 2.59i)17-s + (−1.32 − 2.29i)19-s + (−3.96 + 2.29i)23-s + (−1 + 1.73i)25-s + (1.32 − 2.29i)31-s + (−3.96 + 2.29i)35-s + (−3.5 − 6.06i)37-s + 3.46i·41-s + 9.16i·43-s + (−3.96 − 6.87i)47-s + 7.00·49-s + ⋯
L(s)  = 1  + (0.670 − 0.387i)5-s − 0.999·7-s + (−1.19 − 0.690i)11-s + 0.960i·13-s + (−1.09 − 0.630i)17-s + (−0.303 − 0.525i)19-s + (−0.827 + 0.477i)23-s + (−0.200 + 0.346i)25-s + (0.237 − 0.411i)31-s + (−0.670 + 0.387i)35-s + (−0.575 − 0.996i)37-s + 0.541i·41-s + 1.39i·43-s + (−0.578 − 1.00i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2476878935\)
\(L(\frac12)\) \(\approx\) \(0.2476878935\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T \)
good5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.96 + 2.29i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (4.5 + 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.32 + 2.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.96 - 2.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-1.32 + 2.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 9.16iT - 43T^{2} \)
47 \( 1 + (3.96 + 6.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.96 - 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.96 + 2.29i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.16iT - 71T^{2} \)
73 \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.96 + 2.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.46iT - 97T^{2} \)
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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

−9.423801890902206756870536684151, −9.012940866819939865859100205414, −7.938168923503822329196980092494, −6.89873427578238172091999417391, −6.13501858343338108528619396526, −5.31554391104544217797693165917, −4.27666683240043686262635125860, −3.01945934665557814152385975183, −2.02112644427958919218154372903, −0.10050117155282918327110425986, 2.09701923026291125310806339808, 2.92123545981488831272916035124, 4.13955444668826971207997872736, 5.33877641167049481110865710861, 6.15918770743544933030043835052, 6.84304158859718711240011749337, 7.901360579886362904830349292478, 8.682047818553468861982044978363, 9.842320007172630554727307762236, 10.28279798689886151661312613252

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