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Properties

Label 2-1008-252.31-c1-0-31
Degree $2$
Conductor $1008$
Sign $-0.103 + 0.994i$
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  + (−0.823 + 1.52i)3-s + 0.280i·5-s + (0.164 + 2.64i)7-s + (−1.64 − 2.50i)9-s − 2.82i·11-s + (−4.06 − 2.34i)13-s + (−0.427 − 0.231i)15-s + (−7.00 − 4.04i)17-s + (−0.474 − 0.821i)19-s + (−4.15 − 1.92i)21-s + 0.392i·23-s + 4.92·25-s + (5.17 − 0.440i)27-s + (1.51 + 2.61i)29-s + (1.06 + 1.84i)31-s + ⋯
L(s)  = 1  + (−0.475 + 0.879i)3-s + 0.125i·5-s + (0.0622 + 0.998i)7-s + (−0.548 − 0.836i)9-s − 0.850i·11-s + (−1.12 − 0.651i)13-s + (−0.110 − 0.0596i)15-s + (−1.69 − 0.980i)17-s + (−0.108 − 0.188i)19-s + (−0.907 − 0.419i)21-s + 0.0818i·23-s + 0.984·25-s + (0.996 − 0.0848i)27-s + (0.280 + 0.486i)29-s + (0.191 + 0.330i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.103 + 0.994i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.103 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3699946258\)
\(L(\frac12)\) \(\approx\) \(0.3699946258\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.823 - 1.52i)T \)
7 \( 1 + (-0.164 - 2.64i)T \)
good5 \( 1 - 0.280iT - 5T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (4.06 + 2.34i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (7.00 + 4.04i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.474 + 0.821i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.392iT - 23T^{2} \)
29 \( 1 + (-1.51 - 2.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.06 - 1.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.43 + 4.21i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.478 + 0.276i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.28 - 2.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.39 + 2.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.21 + 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.70 + 9.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.67 + 5.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.85 - 4.53i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.54iT - 71T^{2} \)
73 \( 1 + (0.542 + 0.313i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.3 - 7.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.30 + 9.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-9.90 + 5.71i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.6 - 7.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.691437755486265390268955301133, −8.985790830790191296260464037626, −8.410854290843187967469910444922, −7.02587533474016505241360018574, −6.23215595851922908566165559056, −5.16522905274544587113211249646, −4.78954311079903121694700097675, −3.30649323682296695880361645624, −2.48485348724705586471947799073, −0.17386706875538591429994031698, 1.50257076835765084382925568517, 2.52608957448824599092611483967, 4.31596898801184018612944868072, 4.77665090602305459068376648456, 6.18936644316065944269783342369, 6.90094381205430782681094236250, 7.43612457339019591992413524843, 8.387199714931845515712285207949, 9.341331319888953882778266520289, 10.45792311984094694838918811342

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