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Properties

Label 2-1008-21.17-c3-0-10
Degree $2$
Conductor $1008$
Sign $-0.999 + 0.0354i$
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  + (10.1 + 17.5i)5-s + (−12.0 + 14.0i)7-s + (−15.4 − 8.92i)11-s + 33.1i·13-s + (22.9 − 39.6i)17-s + (35.0 − 20.2i)19-s + (−69.7 + 40.2i)23-s + (−142. + 246. i)25-s + 233. i·29-s + (195. + 112. i)31-s + (−368. − 68.3i)35-s + (135. + 234. i)37-s + 154.·41-s − 367.·43-s + (−263. − 457. i)47-s + ⋯
L(s)  = 1  + (0.905 + 1.56i)5-s + (−0.649 + 0.760i)7-s + (−0.423 − 0.244i)11-s + 0.707i·13-s + (0.326 − 0.566i)17-s + (0.422 − 0.244i)19-s + (−0.632 + 0.365i)23-s + (−1.14 + 1.97i)25-s + 1.49i·29-s + (1.13 + 0.653i)31-s + (−1.78 − 0.329i)35-s + (0.601 + 1.04i)37-s + 0.588·41-s − 1.30·43-s + (−0.818 − 1.41i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0354i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.999 + 0.0354i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ -0.999 + 0.0354i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.400297374\)
\(L(\frac12)\) \(\approx\) \(1.400297374\)
\(L(\frac{5}{2})\) 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 \( 1 + (12.0 - 14.0i)T \)
good5 \( 1 + (-10.1 - 17.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (15.4 + 8.92i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 33.1iT - 2.19e3T^{2} \)
17 \( 1 + (-22.9 + 39.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-35.0 + 20.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (69.7 - 40.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 233. iT - 2.43e4T^{2} \)
31 \( 1 + (-195. - 112. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-135. - 234. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 154.T + 6.89e4T^{2} \)
43 \( 1 + 367.T + 7.95e4T^{2} \)
47 \( 1 + (263. + 457. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (78.8 + 45.5i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (312. - 541. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (78.8 - 45.5i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-431. + 747. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 303. iT - 3.57e5T^{2} \)
73 \( 1 + (999. + 576. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (3.48 + 6.02i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 815.T + 5.71e5T^{2} \)
89 \( 1 + (155. + 269. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.83e3iT - 9.12e5T^{2} \)
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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

−9.939696369675179282319214081694, −9.446776905925142394563223908268, −8.412373052351162715501856964006, −7.20033576276027054558725024443, −6.60603875532187557084438277446, −5.91062478653767023366292039371, −5.00021963456417224615535759117, −3.28514906631008915060324768068, −2.85022948481407598331787150786, −1.74431603761100430087874516399, 0.34300000304872457911850922919, 1.27177480811660621477425077068, 2.53650384261539502522205195776, 3.96943301916252746034159299412, 4.77186764487867519385670019666, 5.76905283466206594367628818431, 6.31978106439247155709353093341, 7.81702008333059663399281809195, 8.188420135222882204023198549710, 9.432799059346628294832429852703

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