Properties

Label 2-1008-21.5-c3-0-9
Degree $2$
Conductor $1008$
Sign $-0.938 + 0.344i$
Analytic cond. $59.4739$
Root an. cond. $7.71193$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.38 + 11.0i)5-s + (−2.53 + 18.3i)7-s + (−46.8 + 27.0i)11-s − 8.85i·13-s + (34.4 + 59.6i)17-s + (141. + 81.9i)19-s + (81.3 + 46.9i)23-s + (−18.9 − 32.8i)25-s + 119. i·29-s + (−85.6 + 49.4i)31-s + (−186. − 145. i)35-s + (−47.0 + 81.5i)37-s − 259.·41-s − 5.01·43-s + (28.6 − 49.6i)47-s + ⋯
L(s)  = 1  + (−0.570 + 0.988i)5-s + (−0.137 + 0.990i)7-s + (−1.28 + 0.741i)11-s − 0.188i·13-s + (0.491 + 0.851i)17-s + (1.71 + 0.989i)19-s + (0.737 + 0.425i)23-s + (−0.151 − 0.262i)25-s + 0.765i·29-s + (−0.496 + 0.286i)31-s + (−0.901 − 0.700i)35-s + (−0.209 + 0.362i)37-s − 0.987·41-s − 0.0177·43-s + (0.0889 − 0.154i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.938 + 0.344i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ -0.938 + 0.344i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.096770343\)
\(L(\frac12)\) \(\approx\) \(1.096770343\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + (2.53 - 18.3i)T \)
good5 \( 1 + (6.38 - 11.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (46.8 - 27.0i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 8.85iT - 2.19e3T^{2} \)
17 \( 1 + (-34.4 - 59.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-141. - 81.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-81.3 - 46.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 119. iT - 2.43e4T^{2} \)
31 \( 1 + (85.6 - 49.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (47.0 - 81.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 259.T + 6.89e4T^{2} \)
43 \( 1 + 5.01T + 7.95e4T^{2} \)
47 \( 1 + (-28.6 + 49.6i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (407. - 235. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (112. + 195. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-370. - 213. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-81.9 - 141. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 79.8iT - 3.57e5T^{2} \)
73 \( 1 + (-666. + 384. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-267. + 463. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 438.T + 5.71e5T^{2} \)
89 \( 1 + (12.8 - 22.2i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.38e3iT - 9.12e5T^{2} \)
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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

−10.13021785834549806949547075950, −9.318603265967590126153698307096, −8.161457956354025989298619044148, −7.59989030656911231762293391715, −6.80471297450835656410305912215, −5.62617739579361155273663759622, −5.06977443670382158057926185364, −3.45103722912472683163817245077, −2.97345615851687255219981545732, −1.66911639679948832155136775931, 0.33905237056715601717056760680, 0.944109884448999755622101263169, 2.78316061049215450515798026442, 3.71560304053695366409960742213, 4.92396300924591266958597548294, 5.27706055327517198158947628896, 6.75290347968669344977047645233, 7.61940664008516694471880957742, 8.124734904164825020524646637902, 9.160027013923307881585877888838

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