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Properties

Label 2-1008-63.59-c1-0-31
Degree $2$
Conductor $1008$
Sign $0.160 + 0.987i$
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  + (−0.478 − 1.66i)3-s + 2.74·5-s + (−1.70 + 2.02i)7-s + (−2.54 + 1.59i)9-s − 0.418i·11-s + (−1.32 − 0.765i)13-s + (−1.31 − 4.56i)15-s + (1.95 − 3.38i)17-s + (5.11 − 2.95i)19-s + (4.18 + 1.86i)21-s − 8.92i·23-s + 2.52·25-s + (3.86 + 3.47i)27-s + (6.00 − 3.46i)29-s + (3.05 − 1.76i)31-s + ⋯
L(s)  = 1  + (−0.276 − 0.961i)3-s + 1.22·5-s + (−0.644 + 0.764i)7-s + (−0.847 + 0.530i)9-s − 0.126i·11-s + (−0.367 − 0.212i)13-s + (−0.338 − 1.17i)15-s + (0.473 − 0.820i)17-s + (1.17 − 0.678i)19-s + (0.913 + 0.407i)21-s − 1.86i·23-s + 0.505·25-s + (0.744 + 0.667i)27-s + (1.11 − 0.643i)29-s + (0.548 − 0.316i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.160 + 0.987i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.160 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.579140262\)
\(L(\frac12)\) \(\approx\) \(1.579140262\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.478 + 1.66i)T \)
7 \( 1 + (1.70 - 2.02i)T \)
good5 \( 1 - 2.74T + 5T^{2} \)
11 \( 1 + 0.418iT - 11T^{2} \)
13 \( 1 + (1.32 + 0.765i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.95 + 3.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.11 + 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.92iT - 23T^{2} \)
29 \( 1 + (-6.00 + 3.46i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.05 + 1.76i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.54 + 7.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.06 - 1.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.77 - 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.885 + 1.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.39 - 1.96i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.02 - 3.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.61 + 0.932i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.38 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.51iT - 71T^{2} \)
73 \( 1 + (-1.65 - 0.952i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.433 - 0.751i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.45 - 5.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.88 - 8.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.200 - 0.115i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.691706930988940025327403519122, −9.050177946272078917241733827230, −8.092811594423497812520218704427, −7.05908483140396213184437854487, −6.30879427377406263948648661058, −5.68270242739235810983557368431, −4.87597308638124164486232106080, −2.82710184214511317716153921744, −2.40441483350191332632833595693, −0.806635659399389031507470686886, 1.40969702484354318975120933106, 3.05940542286613927498769634317, 3.84643393749376822410074296994, 5.09699989249805035155262107650, 5.72847912988435369508853326312, 6.57280489192956173317373786061, 7.57729754021600564736850504302, 8.823532518703448709674947062613, 9.617273674316013850363029246259, 10.14443605176267503329198546716

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