Properties

Label 2-1008-21.5-c3-0-39
Degree $2$
Conductor $1008$
Sign $0.307 + 0.951i$
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  + (−5.42 + 9.40i)5-s + (18.2 + 3.24i)7-s + (44.9 − 25.9i)11-s − 32.1i·13-s + (−40.7 − 70.5i)17-s + (0.0420 + 0.0242i)19-s + (−77.3 − 44.6i)23-s + (3.58 + 6.20i)25-s − 175. i·29-s + (−186. + 107. i)31-s + (−129. + 153. i)35-s + (−32.2 + 55.8i)37-s − 411.·41-s + 234.·43-s + (316. − 547. i)47-s + ⋯
L(s)  = 1  + (−0.485 + 0.840i)5-s + (0.984 + 0.175i)7-s + (1.23 − 0.710i)11-s − 0.686i·13-s + (−0.581 − 1.00i)17-s + (0.000507 + 0.000293i)19-s + (−0.701 − 0.404i)23-s + (0.0286 + 0.0496i)25-s − 1.12i·29-s + (−1.07 + 0.622i)31-s + (−0.625 + 0.742i)35-s + (−0.143 + 0.248i)37-s − 1.56·41-s + 0.832·43-s + (0.980 − 1.69i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.307 + 0.951i$
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: \((2,\ 1008,\ (\ :3/2),\ 0.307 + 0.951i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.679470303\)
\(L(\frac12)\) \(\approx\) \(1.679470303\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.2 - 3.24i)T \)
good5 \( 1 + (5.42 - 9.40i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-44.9 + 25.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 32.1iT - 2.19e3T^{2} \)
17 \( 1 + (40.7 + 70.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-0.0420 - 0.0242i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (77.3 + 44.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 175. iT - 2.43e4T^{2} \)
31 \( 1 + (186. - 107. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (32.2 - 55.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 411.T + 6.89e4T^{2} \)
43 \( 1 - 234.T + 7.95e4T^{2} \)
47 \( 1 + (-316. + 547. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (230. - 132. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-175. - 304. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (673. + 389. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (98.0 + 169. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 142. iT - 3.57e5T^{2} \)
73 \( 1 + (-676. + 390. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-644. + 1.11e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 235.T + 5.71e5T^{2} \)
89 \( 1 + (-335. + 580. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 655. iT - 9.12e5T^{2} \)
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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

−9.284147281884687963807003610025, −8.575091561954410597677007567323, −7.71151035484226829905611261271, −6.95708247229043780933271558673, −6.06607972258588090484469202664, −5.03753004951010503082325084912, −3.98008550191230961782803301233, −3.09774757205995277778859283688, −1.87245469361410310846176952064, −0.43953373218952204793102742408, 1.23790527154334893311560746227, 1.95466454795089383538458624482, 3.91473268038378156524822980173, 4.30423853161970441208145524226, 5.25886499796525547450552116041, 6.42802599543197205090777121632, 7.31121106845929724784126407389, 8.181220648473344252353907371055, 8.895126699488920159928098225282, 9.515390623441768287530737019991

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