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Properties

Label 2-1008-21.2-c0-0-1
Degree $2$
Conductor $1008$
Sign $0.778 - 0.627i$
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  + (1.22 + 0.707i)5-s + (−0.5 + 0.866i)7-s + (1.22 − 0.707i)11-s − 13-s + (−0.5 + 0.866i)19-s + (0.499 + 0.866i)25-s + (−0.5 − 0.866i)31-s + (−1.22 + 0.707i)35-s + (−0.5 + 0.866i)37-s − 1.41i·41-s + 43-s + (1.22 + 0.707i)47-s + (−0.499 − 0.866i)49-s + 2·55-s + (−1.22 − 0.707i)65-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)5-s + (−0.5 + 0.866i)7-s + (1.22 − 0.707i)11-s − 13-s + (−0.5 + 0.866i)19-s + (0.499 + 0.866i)25-s + (−0.5 − 0.866i)31-s + (−1.22 + 0.707i)35-s + (−0.5 + 0.866i)37-s − 1.41i·41-s + 43-s + (1.22 + 0.707i)47-s + (−0.499 − 0.866i)49-s + 2·55-s + (−1.22 − 0.707i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.778 - 0.627i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ 0.778 - 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.191264250\)
\(L(\frac12)\) \(\approx\) \(1.191264250\)
\(L(1)\) 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 \( 1 \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.18436302576464799660853976720, −9.334610416241638717468137318892, −8.981561031132946039111723356607, −7.69070705413337976474501731197, −6.57110503636963350436737277371, −6.10417011737837294407033231399, −5.36211653491241317998011293020, −3.89394717728774835928606380568, −2.77473668505176899484235698189, −1.88924767614950054509550349554, 1.31687132826136465431199494022, 2.50046920687775718428381538131, 4.01334544311742128527241776759, 4.80198187141286266671734113988, 5.79586200211623200220482977968, 6.79799142438297169878400331305, 7.28409990733122891824980565702, 8.734259923796140539001102506325, 9.366778163879547697026004324609, 9.895618501011623695488991868409

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