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Properties

Label 2-1008-63.59-c1-0-27
Degree $2$
Conductor $1008$
Sign $-0.699 + 0.714i$
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  + (−1.64 − 0.550i)3-s − 1.28·5-s + (−1.10 + 2.40i)7-s + (2.39 + 1.80i)9-s − 3.61i·11-s + (3.48 + 2.01i)13-s + (2.11 + 0.708i)15-s + (0.828 − 1.43i)17-s + (−5.15 + 2.97i)19-s + (3.13 − 3.34i)21-s − 0.429i·23-s − 3.34·25-s + (−2.93 − 4.28i)27-s + (6.39 − 3.69i)29-s + (0.841 − 0.485i)31-s + ⋯
L(s)  = 1  + (−0.948 − 0.317i)3-s − 0.575·5-s + (−0.415 + 0.909i)7-s + (0.798 + 0.602i)9-s − 1.09i·11-s + (0.967 + 0.558i)13-s + (0.546 + 0.182i)15-s + (0.200 − 0.347i)17-s + (−1.18 + 0.682i)19-s + (0.683 − 0.730i)21-s − 0.0896i·23-s − 0.668·25-s + (−0.565 − 0.824i)27-s + (1.18 − 0.685i)29-s + (0.151 − 0.0872i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.699 + 0.714i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.699 + 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3650151917\)
\(L(\frac12)\) \(\approx\) \(0.3650151917\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 0.550i)T \)
7 \( 1 + (1.10 - 2.40i)T \)
good5 \( 1 + 1.28T + 5T^{2} \)
11 \( 1 + 3.61iT - 11T^{2} \)
13 \( 1 + (-3.48 - 2.01i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.828 + 1.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.15 - 2.97i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.429iT - 23T^{2} \)
29 \( 1 + (-6.39 + 3.69i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.841 + 0.485i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.16 + 8.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.15 - 8.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.67 + 6.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.01 + 6.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.4 + 6.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.618 - 1.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.75 + 3.32i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.10 + 1.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.66iT - 71T^{2} \)
73 \( 1 + (1.67 + 0.965i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.26 - 3.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.701 + 1.21i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.81 + 8.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.20 - 4.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.762119493779064654458835147903, −8.579426158782945637767064633411, −8.178766799799555531603373426908, −6.87004392209410672673794526029, −6.16754664670675778654678481034, −5.59137911223042644932826055581, −4.37137918978105142461851023366, −3.37479976378601949744999639050, −1.87376296685869515794038728377, −0.20382848312168889387075324986, 1.32081908793534309085180282388, 3.30172655618463146759337486436, 4.24026028363830587139342665480, 4.85846512791671574149280739282, 6.18266268196354839385397332641, 6.76588964216485486008073895611, 7.62406284780858485841244755752, 8.606653324650255541644740029003, 9.715524451301664720610968257139, 10.48670557183810061817392323469

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