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Properties

Label 2-1008-63.25-c1-0-44
Degree $2$
Conductor $1008$
Sign $-0.888 + 0.458i$
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  − 1.73i·3-s + (1.5 − 2.59i)5-s + (2 − 1.73i)7-s − 2.99·9-s + (−1.5 − 2.59i)11-s + (−2.5 − 4.33i)13-s + (−4.5 − 2.59i)15-s + (−1.5 + 2.59i)17-s + (2.5 + 4.33i)19-s + (−2.99 − 3.46i)21-s + (−1.5 + 2.59i)23-s + (−2 − 3.46i)25-s + 5.19i·27-s + (1.5 − 2.59i)29-s + 4·31-s + ⋯
L(s)  = 1  − 0.999i·3-s + (0.670 − 1.16i)5-s + (0.755 − 0.654i)7-s − 0.999·9-s + (−0.452 − 0.783i)11-s + (−0.693 − 1.20i)13-s + (−1.16 − 0.670i)15-s + (−0.363 + 0.630i)17-s + (0.573 + 0.993i)19-s + (−0.654 − 0.755i)21-s + (−0.312 + 0.541i)23-s + (−0.400 − 0.692i)25-s + 0.999i·27-s + (0.278 − 0.482i)29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.888 + 0.458i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.640489022\)
\(L(\frac12)\) \(\approx\) \(1.640489022\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
7 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (-1.5 + 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.619263558612829866938456291901, −8.292487610769629102975359447154, −8.211533024844988270910848156246, −7.28640552385002937919133801497, −5.91556102688506950653119334527, −5.53266593559791639613374244290, −4.48488406827895221351525110804, −2.98385214526719550260434113981, −1.67499453810831390846075392936, −0.75348583842150611768694550139, 2.32138878337855543423008331692, 2.72201376125639236901541679403, 4.37568761540743302055633908492, 4.95060126155134455238889438821, 5.97982032825057503297626016508, 6.90872700278033865757392022791, 7.75988622186440004841201124106, 9.117083148226521486289176463372, 9.389423481983027939106977063499, 10.33798254719065105205939130188

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