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Properties

Label 2-1008-252.187-c1-0-12
Degree $2$
Conductor $1008$
Sign $-0.167 - 0.985i$
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  + (1.72 − 0.108i)3-s + 0.435i·5-s + (−2.49 + 0.891i)7-s + (2.97 − 0.376i)9-s + 4.82i·11-s + (−4.31 + 2.49i)13-s + (0.0473 + 0.752i)15-s + (−4.92 + 2.84i)17-s + (−3.70 + 6.42i)19-s + (−4.20 + 1.81i)21-s − 3.16i·23-s + 4.81·25-s + (5.10 − 0.974i)27-s + (2.49 − 4.32i)29-s + (1.32 − 2.30i)31-s + ⋯
L(s)  = 1  + (0.998 − 0.0628i)3-s + 0.194i·5-s + (−0.941 + 0.337i)7-s + (0.992 − 0.125i)9-s + 1.45i·11-s + (−1.19 + 0.691i)13-s + (0.0122 + 0.194i)15-s + (−1.19 + 0.689i)17-s + (−0.851 + 1.47i)19-s + (−0.918 + 0.395i)21-s − 0.658i·23-s + 0.962·25-s + (0.982 − 0.187i)27-s + (0.463 − 0.802i)29-s + (0.238 − 0.413i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.167 - 0.985i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.167 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.553492598\)
\(L(\frac12)\) \(\approx\) \(1.553492598\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.108i)T \)
7 \( 1 + (2.49 - 0.891i)T \)
good5 \( 1 - 0.435iT - 5T^{2} \)
11 \( 1 - 4.82iT - 11T^{2} \)
13 \( 1 + (4.31 - 2.49i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.92 - 2.84i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.70 - 6.42i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.16iT - 23T^{2} \)
29 \( 1 + (-2.49 + 4.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.32 + 2.30i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.06 + 1.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.112 + 0.0650i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.53 - 3.77i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.29 - 7.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.60 + 9.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.20 - 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.65 + 4.42i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.811 - 0.468i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.27iT - 71T^{2} \)
73 \( 1 + (-11.2 + 6.48i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.98 - 1.72i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.68 + 6.38i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (13.2 + 7.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.83 + 1.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.924932523267726629136566787213, −9.468507512827079394774681143568, −8.613008028109480945517402157100, −7.69694171653545515458814468634, −6.84060504585646166945617797632, −6.26979029370775484479710638064, −4.58174987154155678833221094116, −4.05067980967400903272488925863, −2.60214819140874518825280670404, −2.04449939890005357081380413636, 0.59019925792482768059002110831, 2.60026992035926355470873853830, 3.10754053004774295464536344156, 4.31794140798068870306600236828, 5.25563443935245905545416164796, 6.66302786865743578234864757706, 7.12089373967582856418113858946, 8.259480072821472443485704816814, 8.983680982006736801496390456988, 9.470504528485608009819404427726

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