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Properties

Label 2-1008-63.58-c1-0-14
Degree $2$
Conductor $1008$
Sign $0.995 - 0.0984i$
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  + (−1.65 − 0.518i)3-s + (−0.712 − 1.23i)5-s + (2.36 − 1.19i)7-s + (2.46 + 1.71i)9-s + (−2.46 + 4.27i)11-s + (−1.37 + 2.38i)13-s + (0.537 + 2.40i)15-s + (0.559 + 0.969i)17-s + (2.00 − 3.47i)19-s + (−4.52 + 0.751i)21-s + (2.71 + 4.70i)23-s + (1.48 − 2.57i)25-s + (−3.18 − 4.10i)27-s + (3.40 + 5.89i)29-s − 2.50·31-s + ⋯
L(s)  = 1  + (−0.954 − 0.299i)3-s + (−0.318 − 0.551i)5-s + (0.892 − 0.451i)7-s + (0.820 + 0.571i)9-s + (−0.743 + 1.28i)11-s + (−0.381 + 0.661i)13-s + (0.138 + 0.621i)15-s + (0.135 + 0.235i)17-s + (0.460 − 0.797i)19-s + (−0.986 + 0.163i)21-s + (0.566 + 0.981i)23-s + (0.296 − 0.514i)25-s + (−0.612 − 0.790i)27-s + (0.632 + 1.09i)29-s − 0.450·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.995 - 0.0984i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.995 - 0.0984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.113297226\)
\(L(\frac12)\) \(\approx\) \(1.113297226\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 + 0.518i)T \)
7 \( 1 + (-2.36 + 1.19i)T \)
good5 \( 1 + (0.712 + 1.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.46 - 4.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.37 - 2.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.559 - 0.969i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.00 + 3.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.71 - 4.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.40 - 5.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.50T + 31T^{2} \)
37 \( 1 + (-0.709 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.124 + 0.215i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.498 - 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.47T + 47T^{2} \)
53 \( 1 + (0.410 + 0.710i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.58T + 59T^{2} \)
61 \( 1 - 0.0752T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 0.0804T + 71T^{2} \)
73 \( 1 + (-5.34 - 9.25i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 1.84T + 79T^{2} \)
83 \( 1 + (-7.23 - 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.76 + 11.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.70 - 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.13409503618327108311366356252, −9.219679213738750636715650334490, −8.111943900195589260284073602194, −7.29888202560490801255423575045, −6.87246280656418902715996795170, −5.31073421336055456737731668437, −4.92784963080323196910357227432, −4.11353784505149525956489295612, −2.20793891611074692591683017340, −1.03163659746796051064322069071, 0.75938539082710144559210803111, 2.60880612043996317595819748511, 3.71129444186013015595174694017, 4.97065350159684916268840708854, 5.53150881122014239097244514116, 6.38054060291110193858044896795, 7.51959697534113407545574018011, 8.143617035339844584648744170686, 9.161116423712901218625910831068, 10.33025480575122392958727079493

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