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Properties

Label 2-1008-63.25-c1-0-32
Degree $2$
Conductor $1008$
Sign $-0.759 + 0.650i$
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  + (−1.71 − 0.210i)3-s + (0.0309 − 0.0536i)5-s + (0.981 − 2.45i)7-s + (2.91 + 0.724i)9-s + (−1.59 − 2.75i)11-s + (−0.252 − 0.437i)13-s + (−0.0645 + 0.0857i)15-s + (−0.554 + 0.960i)17-s + (−0.933 − 1.61i)19-s + (−2.20 + 4.01i)21-s + (−3.10 + 5.37i)23-s + (2.49 + 4.32i)25-s + (−4.85 − 1.85i)27-s + (2.39 − 4.15i)29-s + 2.53·31-s + ⋯
L(s)  = 1  + (−0.992 − 0.121i)3-s + (0.0138 − 0.0240i)5-s + (0.371 − 0.928i)7-s + (0.970 + 0.241i)9-s + (−0.479 − 0.830i)11-s + (−0.0700 − 0.121i)13-s + (−0.0166 + 0.0221i)15-s + (−0.134 + 0.233i)17-s + (−0.214 − 0.370i)19-s + (−0.481 + 0.876i)21-s + (−0.646 + 1.12i)23-s + (0.499 + 0.865i)25-s + (−0.933 − 0.357i)27-s + (0.445 − 0.770i)29-s + 0.455·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.759 + 0.650i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.759 + 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6635599841\)
\(L(\frac12)\) \(\approx\) \(0.6635599841\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.210i)T \)
7 \( 1 + (-0.981 + 2.45i)T \)
good5 \( 1 + (-0.0309 + 0.0536i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.59 + 2.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.252 + 0.437i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.554 - 0.960i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.933 + 1.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.10 - 5.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.39 + 4.15i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.53T + 31T^{2} \)
37 \( 1 + (4.26 + 7.38i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.94 + 8.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.95 + 6.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.58T + 47T^{2} \)
53 \( 1 + (1.58 - 2.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.01T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + 3.33T + 67T^{2} \)
71 \( 1 + 2.25T + 71T^{2} \)
73 \( 1 + (-2.07 + 3.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.97T + 79T^{2} \)
83 \( 1 + (2.17 - 3.76i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.30 + 7.44i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.27 - 5.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.899129753514824825357856346501, −8.797600168315796387885420762108, −7.72664728588378060026906053793, −7.17740294494994253265493893392, −6.12685767771893018040643458199, −5.37096039206925451447906353240, −4.45658429846909683692840734830, −3.44351691719679793591011860966, −1.71298817592970080296122480046, −0.34721212452995484602252144035, 1.62554830204700901353058747367, 2.88422099805416727075631662959, 4.65300723724097111987029213685, 4.83115611183643307653791767397, 6.12864797533027619939349293396, 6.62256730149167675230934145083, 7.81187269745823488495450214746, 8.588869178092959426325686672548, 9.688140441077272834158311860526, 10.29541737997583355569278538387

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