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Properties

Label 2-1008-63.4-c1-0-4
Degree $2$
Conductor $1008$
Sign $-0.990 - 0.139i$
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  + (0.849 + 1.50i)3-s − 1.58·5-s + (1.80 − 1.93i)7-s + (−1.55 + 2.56i)9-s − 5.17·11-s + (−0.681 + 1.18i)13-s + (−1.34 − 2.38i)15-s + (−2.30 + 3.99i)17-s + (−0.0321 − 0.0557i)19-s + (4.45 + 1.08i)21-s − 6.74·23-s − 2.49·25-s + (−5.19 − 0.166i)27-s + (4.70 + 8.15i)29-s + (−1.33 − 2.30i)31-s + ⋯
L(s)  = 1  + (0.490 + 0.871i)3-s − 0.707·5-s + (0.683 − 0.729i)7-s + (−0.518 + 0.855i)9-s − 1.55·11-s + (−0.189 + 0.327i)13-s + (−0.347 − 0.616i)15-s + (−0.559 + 0.969i)17-s + (−0.00738 − 0.0127i)19-s + (0.971 + 0.237i)21-s − 1.40·23-s − 0.499·25-s + (−0.999 − 0.0320i)27-s + (0.874 + 1.51i)29-s + (−0.239 − 0.414i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.990 - 0.139i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.990 - 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6439838734\)
\(L(\frac12)\) \(\approx\) \(0.6439838734\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.849 - 1.50i)T \)
7 \( 1 + (-1.80 + 1.93i)T \)
good5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 + 5.17T + 11T^{2} \)
13 \( 1 + (0.681 - 1.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.30 - 3.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0321 + 0.0557i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.74T + 23T^{2} \)
29 \( 1 + (-4.70 - 8.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.33 + 2.30i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.880 - 1.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.858 - 1.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.12 - 8.86i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.60 + 4.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.479 - 0.831i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.66 + 8.08i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.19 - 12.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.24 + 10.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.49T + 71T^{2} \)
73 \( 1 + (0.941 - 1.63i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.26 + 5.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.08 - 8.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.12 + 7.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.26 + 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.54652661780225175521422716191, −9.644998553709095638088329464617, −8.478652040468236006091059719980, −8.009042415462320301801001339077, −7.37632057259322832685680362752, −5.94250747443379039416959836217, −4.80090122834415215419191974484, −4.27853677534534323959710468689, −3.28395287578745983222642671704, −2.03610031275988853214325660222, 0.25365790897046320262452097475, 2.15482057579709167463424266824, 2.79827036492111633824703107145, 4.19854616858356544639153328072, 5.30995794071753632587115421689, 6.14063548765820543419282610817, 7.44464815169353862087374071491, 7.84809968572846771933025342581, 8.445924535595558089214261227571, 9.388231723549936979043823885270

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