Properties

Label 2-1008-63.25-c1-0-35
Degree $2$
Conductor $1008$
Sign $0.367 + 0.930i$
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  + (0.221 + 1.71i)3-s + (1.84 − 3.19i)5-s + (−0.926 − 2.47i)7-s + (−2.90 + 0.760i)9-s + (−0.446 − 0.772i)11-s + (0.598 + 1.03i)13-s + (5.90 + 2.46i)15-s + (−0.124 + 0.216i)17-s + (−1.40 − 2.43i)19-s + (4.05 − 2.14i)21-s + (1.23 − 2.14i)23-s + (−4.31 − 7.47i)25-s + (−1.94 − 4.81i)27-s + (2.07 − 3.58i)29-s − 3.58·31-s + ⋯
L(s)  = 1  + (0.127 + 0.991i)3-s + (0.825 − 1.43i)5-s + (−0.350 − 0.936i)7-s + (−0.967 + 0.253i)9-s + (−0.134 − 0.233i)11-s + (0.165 + 0.287i)13-s + (1.52 + 0.636i)15-s + (−0.0303 + 0.0525i)17-s + (−0.322 − 0.557i)19-s + (0.884 − 0.467i)21-s + (0.258 − 0.447i)23-s + (−0.863 − 1.49i)25-s + (−0.374 − 0.927i)27-s + (0.384 − 0.666i)29-s − 0.643·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.367 + 0.930i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.367 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.535525197\)
\(L(\frac12)\) \(\approx\) \(1.535525197\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.221 - 1.71i)T \)
7 \( 1 + (0.926 + 2.47i)T \)
good5 \( 1 + (-1.84 + 3.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.446 + 0.772i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.598 - 1.03i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.124 - 0.216i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.40 + 2.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.23 + 2.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.07 + 3.58i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 + (2.36 + 4.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.39 + 4.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.98 + 8.64i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + (4.94 - 8.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.81T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + 1.02T + 67T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 + (0.915 - 1.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.79T + 79T^{2} \)
83 \( 1 + (6.16 - 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.20 + 2.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.52 + 9.56i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.760459868460895196935469954817, −8.970035305026580709816571511881, −8.599019909531672816658420531833, −7.36040091248780279350114450587, −6.12781368440311092597474358662, −5.33699547899958000274851356759, −4.49690775582698544181416234538, −3.78726887476274525379073021608, −2.30050884823586810539645575230, −0.67432416799957010167286783723, 1.75448173981580667955203951564, 2.65998822119982972169648103332, 3.35611019471545829189951929903, 5.32185889790307047961925123231, 6.09206626922651715644330360287, 6.64600939545751716730388530256, 7.44932184920824980021320312166, 8.399725791219529256476938111144, 9.318558044346840042123077999161, 10.10613034173045721779826003233

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