Properties

Label 2-1008-63.4-c1-0-12
Degree $2$
Conductor $1008$
Sign $-0.205 - 0.978i$
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  + (−1.34 + 1.09i)3-s + 2.66·5-s + (−1.94 − 1.79i)7-s + (0.613 − 2.93i)9-s − 1.36·11-s + (−2.75 + 4.77i)13-s + (−3.58 + 2.91i)15-s + (−1.23 + 2.14i)17-s + (2.19 + 3.80i)19-s + (4.57 + 0.286i)21-s + 4.69·23-s + 2.11·25-s + (2.38 + 4.61i)27-s + (2.94 + 5.10i)29-s + (1.55 + 2.69i)31-s + ⋯
L(s)  = 1  + (−0.776 + 0.630i)3-s + 1.19·5-s + (−0.735 − 0.678i)7-s + (0.204 − 0.978i)9-s − 0.411·11-s + (−0.764 + 1.32i)13-s + (−0.925 + 0.752i)15-s + (−0.300 + 0.520i)17-s + (0.503 + 0.872i)19-s + (0.998 + 0.0626i)21-s + 0.977·23-s + 0.423·25-s + (0.458 + 0.888i)27-s + (0.547 + 0.948i)29-s + (0.279 + 0.484i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.205 - 0.978i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.056894863\)
\(L(\frac12)\) \(\approx\) \(1.056894863\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.34 - 1.09i)T \)
7 \( 1 + (1.94 + 1.79i)T \)
good5 \( 1 - 2.66T + 5T^{2} \)
11 \( 1 + 1.36T + 11T^{2} \)
13 \( 1 + (2.75 - 4.77i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.23 - 2.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.19 - 3.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.69T + 23T^{2} \)
29 \( 1 + (-2.94 - 5.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.55 - 2.69i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.15 + 5.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.38 + 2.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.87 - 8.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.02 - 8.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.47 - 2.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.77 - 3.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.663 - 1.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.14 - 7.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (1.11 - 1.93i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.41 + 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.15 + 8.93i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.73 - 13.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.55 + 4.42i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.26478481602770702716611442730, −9.465390802447161860063679397209, −9.033081223341974866825396092663, −7.41262719015771162454978014576, −6.60710082422402774527677629523, −5.96801517262218169928252493983, −5.02092857917174872160360864855, −4.16641578888797375645638465866, −2.94725228922059478794676551909, −1.42777557617215872484774865101, 0.54054385281397182653470729561, 2.24554840474941599851314923060, 2.91704819866548416530125725827, 5.01779305769452455836973224709, 5.39383115931040905891493368793, 6.28008438218512975643720623518, 6.98423213933501419704671386542, 7.948310380392899152075434847544, 9.041373680992686925692589536212, 9.916331132811192758852380889802

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