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Properties

Label 2-1008-48.11-c1-0-37
Degree $2$
Conductor $1008$
Sign $-0.211 + 0.977i$
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  + (−0.153 + 1.40i)2-s + (−1.95 − 0.432i)4-s + (0.0893 − 0.0893i)5-s + 7-s + (0.907 − 2.67i)8-s + (0.111 + 0.139i)10-s + (−2.42 − 2.42i)11-s + (−3.86 + 3.86i)13-s + (−0.153 + 1.40i)14-s + (3.62 + 1.68i)16-s + 0.794i·17-s + (−2.65 − 2.65i)19-s + (−0.213 + 0.135i)20-s + (3.78 − 3.03i)22-s − 3.92i·23-s + ⋯
L(s)  = 1  + (−0.108 + 0.994i)2-s + (−0.976 − 0.216i)4-s + (0.0399 − 0.0399i)5-s + 0.377·7-s + (0.320 − 0.947i)8-s + (0.0353 + 0.0440i)10-s + (−0.731 − 0.731i)11-s + (−1.07 + 1.07i)13-s + (−0.0410 + 0.375i)14-s + (0.906 + 0.421i)16-s + 0.192i·17-s + (−0.608 − 0.608i)19-s + (−0.0476 + 0.0303i)20-s + (0.806 − 0.647i)22-s − 0.817i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.211 + 0.977i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.211 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1680305042\)
\(L(\frac12)\) \(\approx\) \(0.1680305042\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.153 - 1.40i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (-0.0893 + 0.0893i)T - 5iT^{2} \)
11 \( 1 + (2.42 + 2.42i)T + 11iT^{2} \)
13 \( 1 + (3.86 - 3.86i)T - 13iT^{2} \)
17 \( 1 - 0.794iT - 17T^{2} \)
19 \( 1 + (2.65 + 2.65i)T + 19iT^{2} \)
23 \( 1 + 3.92iT - 23T^{2} \)
29 \( 1 + (7.47 + 7.47i)T + 29iT^{2} \)
31 \( 1 - 5.55iT - 31T^{2} \)
37 \( 1 + (6.35 + 6.35i)T + 37iT^{2} \)
41 \( 1 + 6.96T + 41T^{2} \)
43 \( 1 + (1.25 - 1.25i)T - 43iT^{2} \)
47 \( 1 + 6.48T + 47T^{2} \)
53 \( 1 + (-0.620 + 0.620i)T - 53iT^{2} \)
59 \( 1 + (-3.39 - 3.39i)T + 59iT^{2} \)
61 \( 1 + (-7.51 + 7.51i)T - 61iT^{2} \)
67 \( 1 + (-2.22 - 2.22i)T + 67iT^{2} \)
71 \( 1 + 7.95iT - 71T^{2} \)
73 \( 1 + 12.9iT - 73T^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 + (-11.2 + 11.2i)T - 83iT^{2} \)
89 \( 1 + 5.52T + 89T^{2} \)
97 \( 1 + 4.33T + 97T^{2} \)
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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

−9.475275128480174140318041465678, −8.796767345979595829026465615258, −8.012217683678556058757351245430, −7.18879398886456999266755086293, −6.45370417950326772004596050432, −5.36262559298814951567631414324, −4.76070206038882765425572902400, −3.64619290791920212648718901702, −2.04777920678489179475006089479, −0.07581552038138815053019516461, 1.75078028999234313490235764499, 2.71405929474582752352644167756, 3.81625791631334061407774277128, 4.98555771490796125947175292274, 5.45155900849541897990723301985, 7.08280643990619467896691209880, 7.917375634645918808926034610713, 8.552785824382580975585622326284, 9.814852682653267200155570045471, 10.07560195071189125438516803529

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