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Properties

Label 2-1008-252.247-c0-0-0
Degree $2$
Conductor $1008$
Sign $-0.235 - 0.971i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (0.5 + 0.866i)5-s + i·7-s − 9-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)13-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s − 21-s + (−0.866 + 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)33-s + (−0.866 + 0.5i)35-s + ⋯
L(s)  = 1  + i·3-s + (0.5 + 0.866i)5-s + i·7-s − 9-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)13-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s − 21-s + (−0.866 + 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)33-s + (−0.866 + 0.5i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ -0.235 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.085017762\)
\(L(\frac12)\) \(\approx\) \(1.085017762\)
\(L(1)\) 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 \( 1 - iT \)
7 \( 1 - iT \)
good5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.39492243333901460866245774480, −9.539265252782859095304439085291, −9.007689052168776707306606674689, −8.132344519781096156026467048270, −6.79481032792170685154119896306, −6.09342710487569622564087794698, −5.26142373783177917735640392806, −4.22031365536126101856571392207, −3.09115664224246892900034531127, −2.26872448221276452197948913620, 1.13127759695365024192134292547, 2.03803757474575597421222100435, 3.76666502416412601559604205579, 4.54896543937112588869886783400, 6.09166861796366737010755714980, 6.31391416565354780231513837545, 7.38240148728258592432625179078, 8.479138322119725953995842423427, 8.750542815617534149886073217965, 9.892302087129899256736961168717

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