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Properties

Label 2-1008-112.67-c0-0-0
Degree $2$
Conductor $1008$
Sign $0.657 - 0.753i$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.965 + 0.258i)5-s + (−0.5 + 0.866i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)10-s + (−0.965 + 0.258i)11-s + (1 + i)13-s + (0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (0.707 + 0.707i)20-s + 22-s + (0.707 + 1.22i)23-s + (−0.707 − 1.22i)26-s + (−0.866 + 0.5i)28-s + (−0.707 − 0.707i)29-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.965 + 0.258i)5-s + (−0.5 + 0.866i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)10-s + (−0.965 + 0.258i)11-s + (1 + i)13-s + (0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (0.707 + 0.707i)20-s + 22-s + (0.707 + 1.22i)23-s + (−0.707 − 1.22i)26-s + (−0.866 + 0.5i)28-s + (−0.707 − 0.707i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.657 - 0.753i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ 0.657 - 0.753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7157801155\)
\(L(\frac12)\) \(\approx\) \(0.7157801155\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
31 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.985592725342030340083563217147, −9.511733670035445986297031630081, −8.885615450769485776221014680410, −7.931102597508668664517571540607, −6.95571440116968898236618589441, −6.11452448772899513878614151422, −5.45832197819444859949787335577, −3.71478167488610124515557801803, −2.56618236682001401198226931814, −1.78492534386597582453196419161, 0.946475793828449386761839104308, 2.38746767481844415820798082049, 3.54270627502416413623786514910, 5.27292962807089629284061540506, 5.83220545559007205584739237763, 6.81481044329021410610316900629, 7.57843894342348254062225155586, 8.551045969787259273174510997097, 9.146812374051761625929672857592, 10.19008546488485668249032177067

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