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Properties

Label 2-1008-63.16-c1-0-7
Degree $2$
Conductor $1008$
Sign $-0.259 - 0.965i$
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  + (−1.51 + 0.831i)3-s − 2.52·5-s + (1.07 + 2.41i)7-s + (1.61 − 2.52i)9-s + 5.71·11-s + (−2.45 − 4.24i)13-s + (3.83 − 2.09i)15-s + (2.49 + 4.32i)17-s + (0.00383 − 0.00664i)19-s + (−3.64 − 2.77i)21-s − 0.667·23-s + 1.36·25-s + (−0.355 + 5.18i)27-s + (3.85 − 6.66i)29-s + (−3.88 + 6.72i)31-s + ⋯
L(s)  = 1  + (−0.877 + 0.480i)3-s − 1.12·5-s + (0.407 + 0.913i)7-s + (0.539 − 0.842i)9-s + 1.72·11-s + (−0.680 − 1.17i)13-s + (0.989 − 0.541i)15-s + (0.605 + 1.04i)17-s + (0.000880 − 0.00152i)19-s + (−0.795 − 0.605i)21-s − 0.139·23-s + 0.273·25-s + (−0.0685 + 0.997i)27-s + (0.715 − 1.23i)29-s + (−0.697 + 1.20i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.259 - 0.965i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.259 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8523664066\)
\(L(\frac12)\) \(\approx\) \(0.8523664066\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 - 0.831i)T \)
7 \( 1 + (-1.07 - 2.41i)T \)
good5 \( 1 + 2.52T + 5T^{2} \)
11 \( 1 - 5.71T + 11T^{2} \)
13 \( 1 + (2.45 + 4.24i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.49 - 4.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.00383 + 0.00664i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.667T + 23T^{2} \)
29 \( 1 + (-3.85 + 6.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.88 - 6.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.19 - 5.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.21 - 9.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.42 - 7.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.08 - 1.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.69 + 6.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.261 - 0.453i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.49 - 7.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.54 - 4.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.68T + 71T^{2} \)
73 \( 1 + (1.52 + 2.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.08 - 5.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.258 + 0.448i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.19 + 2.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.32 + 7.49i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.21861228246434719227555534317, −9.503549523269254341101660847126, −8.464249425601558984012805183864, −7.83614416528461598672403776466, −6.64745416182787457995956693246, −5.92116940352598773925803655266, −4.93255440294755515971928959330, −4.09913554373924184817110530280, −3.19586282142486625813814375472, −1.24811961390522762843430877766, 0.51782402093774804472502658207, 1.78072025529532519721707470601, 3.77300458092038390933106682166, 4.32006962470879741085372824026, 5.29723207041760918317691401287, 6.65672266406090694817641369450, 7.15354611824256704067720793797, 7.68405002577135623114621325013, 8.908685940510840261049112234628, 9.746583159249196633758978260758

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