Properties

Label 2-1008-63.25-c1-0-6
Degree $2$
Conductor $1008$
Sign $-0.310 - 0.950i$
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  + (−1.04 + 1.38i)3-s + (1.33 − 2.31i)5-s + (−2.54 + 0.714i)7-s + (−0.829 − 2.88i)9-s + (−1.99 − 3.45i)11-s + (1.00 + 1.73i)13-s + (1.80 + 4.25i)15-s + (−3.57 + 6.18i)17-s + (4.01 + 6.96i)19-s + (1.66 − 4.26i)21-s + (−0.443 + 0.768i)23-s + (−1.06 − 1.83i)25-s + (4.85 + 1.85i)27-s + (−1.35 + 2.33i)29-s + 1.22·31-s + ⋯
L(s)  = 1  + (−0.601 + 0.798i)3-s + (0.596 − 1.03i)5-s + (−0.962 + 0.270i)7-s + (−0.276 − 0.961i)9-s + (−0.600 − 1.04i)11-s + (0.277 + 0.480i)13-s + (0.466 + 1.09i)15-s + (−0.866 + 1.50i)17-s + (0.922 + 1.59i)19-s + (0.363 − 0.931i)21-s + (−0.0925 + 0.160i)23-s + (−0.212 − 0.367i)25-s + (0.934 + 0.357i)27-s + (−0.250 + 0.434i)29-s + 0.220·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.310 - 0.950i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.310 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8200673231\)
\(L(\frac12)\) \(\approx\) \(0.8200673231\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.04 - 1.38i)T \)
7 \( 1 + (2.54 - 0.714i)T \)
good5 \( 1 + (-1.33 + 2.31i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.99 + 3.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.00 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.57 - 6.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.01 - 6.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.443 - 0.768i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.35 - 2.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.22T + 31T^{2} \)
37 \( 1 + (-5.26 - 9.11i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.43 + 2.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.40 - 5.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + (2.38 - 4.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.58T + 59T^{2} \)
61 \( 1 + 9.49T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 4.62T + 71T^{2} \)
73 \( 1 + (-2.01 + 3.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.02T + 79T^{2} \)
83 \( 1 + (-5.26 + 9.12i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.72 - 2.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.12 - 1.94i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.23629644283694350725602245401, −9.323654445334263456546026259155, −8.877787007475380409472264100172, −7.949291481066446818656416310665, −6.19906415091253923632833879321, −6.05787270652617060283066295914, −5.12720672064773948781476222453, −4.05655430966243112151617583183, −3.14585024741629727450722475018, −1.35167304499001374035748412469, 0.41746024029951294531747402240, 2.37255347923322185807141522890, 2.87944157838417146523503962683, 4.61346634493028608487961753151, 5.59021919060013808723991865363, 6.49021297641680066782442160709, 7.15632320852931198774874930088, 7.51388034944718243839448658471, 9.110776787679688449377678727779, 9.797392669227040456493056155805

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