Properties

Label 2-1008-21.5-c3-0-11
Degree $2$
Conductor $1008$
Sign $-0.0818 - 0.996i$
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  + (−4.27 + 7.39i)5-s + (12.5 − 13.5i)7-s + (−27.4 + 15.8i)11-s + 9.92i·13-s + (−63.7 − 110. i)17-s + (100. + 58.2i)19-s + (55.8 + 32.2i)23-s + (26.0 + 45.0i)25-s − 113. i·29-s + (−6.33 + 3.65i)31-s + (46.8 + 151. i)35-s + (−184. + 319. i)37-s − 211.·41-s + 432.·43-s + (−200. + 346. i)47-s + ⋯
L(s)  = 1  + (−0.382 + 0.661i)5-s + (0.679 − 0.733i)7-s + (−0.752 + 0.434i)11-s + 0.211i·13-s + (−0.909 − 1.57i)17-s + (1.21 + 0.703i)19-s + (0.506 + 0.292i)23-s + (0.208 + 0.360i)25-s − 0.723i·29-s + (−0.0367 + 0.0211i)31-s + (0.226 + 0.729i)35-s + (−0.820 + 1.42i)37-s − 0.807·41-s + 1.53·43-s + (−0.620 + 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0818 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0818 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0818 - 0.996i$
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: \((2,\ 1008,\ (\ :3/2),\ -0.0818 - 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.386039259\)
\(L(\frac12)\) \(\approx\) \(1.386039259\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.5 + 13.5i)T \)
good5 \( 1 + (4.27 - 7.39i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (27.4 - 15.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 9.92iT - 2.19e3T^{2} \)
17 \( 1 + (63.7 + 110. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-100. - 58.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-55.8 - 32.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 113. iT - 2.43e4T^{2} \)
31 \( 1 + (6.33 - 3.65i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (184. - 319. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 211.T + 6.89e4T^{2} \)
43 \( 1 - 432.T + 7.95e4T^{2} \)
47 \( 1 + (200. - 346. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-121. + 70.3i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-259. - 449. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-23.5 - 13.6i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (68.3 + 118. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 604. iT - 3.57e5T^{2} \)
73 \( 1 + (41.9 - 24.2i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (415. - 719. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 37.2T + 5.71e5T^{2} \)
89 \( 1 + (235. - 407. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 522. iT - 9.12e5T^{2} \)
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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

−9.911018979817800244384202024950, −9.025614472322328652146418315148, −7.82663975372402623851413568610, −7.39510019939233955997983988316, −6.68328489412731174658450084415, −5.27852476553272016386054482382, −4.62768710753190703415522682999, −3.47168544890130885050137026217, −2.49633970630991971043207702490, −1.08748203715038715409827658256, 0.38480929888588060042711246001, 1.73371332019896879629343539303, 2.89199314894344577176325892676, 4.13032920989154581468563590803, 5.11006014488457110029607761064, 5.66681346359956993772463695281, 6.89772510117680725031825917192, 7.899932942117326617201775135038, 8.634283528173216553740243110381, 8.998572278820343886250252460457

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