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Properties

Label 2-1008-21.5-c3-0-25
Degree $2$
Conductor $1008$
Sign $0.758 + 0.651i$
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  + (5.74 − 9.94i)5-s + (−17.2 + 6.73i)7-s + (−49.6 + 28.6i)11-s − 9.20i·13-s + (14.5 + 25.1i)17-s + (32.6 + 18.8i)19-s + (7.27 + 4.19i)23-s + (−3.41 − 5.91i)25-s + 62.3i·29-s + (48.8 − 28.2i)31-s + (−32.1 + 210. i)35-s + (146. − 252. i)37-s + 54.2·41-s + 438.·43-s + (128. − 222. i)47-s + ⋯
L(s)  = 1  + (0.513 − 0.889i)5-s + (−0.931 + 0.363i)7-s + (−1.36 + 0.785i)11-s − 0.196i·13-s + (0.207 + 0.359i)17-s + (0.393 + 0.227i)19-s + (0.0659 + 0.0380i)23-s + (−0.0273 − 0.0473i)25-s + 0.399i·29-s + (0.283 − 0.163i)31-s + (−0.155 + 1.01i)35-s + (0.649 − 1.12i)37-s + 0.206·41-s + 1.55·43-s + (0.398 − 0.690i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.758 + 0.651i$
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: \((2,\ 1008,\ (\ :3/2),\ 0.758 + 0.651i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.612393367\)
\(L(\frac12)\) \(\approx\) \(1.612393367\)
\(L(\frac{5}{2})\) 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 \( 1 + (17.2 - 6.73i)T \)
good5 \( 1 + (-5.74 + 9.94i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (49.6 - 28.6i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 9.20iT - 2.19e3T^{2} \)
17 \( 1 + (-14.5 - 25.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-32.6 - 18.8i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-7.27 - 4.19i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 62.3iT - 2.43e4T^{2} \)
31 \( 1 + (-48.8 + 28.2i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-146. + 252. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 54.2T + 6.89e4T^{2} \)
43 \( 1 - 438.T + 7.95e4T^{2} \)
47 \( 1 + (-128. + 222. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-515. + 297. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (238. + 412. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (548. + 316. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-308. - 534. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 396. iT - 3.57e5T^{2} \)
73 \( 1 + (-39.8 + 23.0i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (344. - 597. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.28e3T + 5.71e5T^{2} \)
89 \( 1 + (-595. + 1.03e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 946. iT - 9.12e5T^{2} \)
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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

−9.537941473046269703734499660842, −8.790355980097552482723005528013, −7.84595343492860090530604213368, −7.01683141940524089907685278490, −5.76424868246992848743737749575, −5.37633729364096988667483979704, −4.26244006705662054523855862815, −2.97991912927527820339213195770, −1.99922509970417993163972787514, −0.55774948660893897029286078165, 0.77876187947635142697833357725, 2.66010737314629875193537955264, 2.98138655212733564562361192410, 4.33148041440421658497669128660, 5.61937908389436244685418114887, 6.21646495190686978425563512115, 7.14732253788056678609288823592, 7.84487520596530390907144777799, 8.985849123308330691290348394437, 9.829873336438242235005940196929

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