Properties

Label 2-1008-21.17-c3-0-29
Degree $2$
Conductor $1008$
Sign $0.749 + 0.662i$
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  + (3.30 + 5.73i)5-s + (4.31 − 18.0i)7-s + (1.54 + 0.893i)11-s + 4.72i·13-s + (−23.8 + 41.3i)17-s + (40.1 − 23.1i)19-s + (30.2 − 17.4i)23-s + (40.5 − 70.3i)25-s + 48.3i·29-s + (107. + 62.2i)31-s + (117. − 34.8i)35-s + (−137. − 238. i)37-s − 37.3·41-s + 215.·43-s + (−53.1 − 91.9i)47-s + ⋯
L(s)  = 1  + (0.295 + 0.512i)5-s + (0.233 − 0.972i)7-s + (0.0424 + 0.0244i)11-s + 0.100i·13-s + (−0.340 + 0.589i)17-s + (0.484 − 0.279i)19-s + (0.273 − 0.158i)23-s + (0.324 − 0.562i)25-s + 0.309i·29-s + (0.624 + 0.360i)31-s + (0.567 − 0.168i)35-s + (−0.612 − 1.06i)37-s − 0.142·41-s + 0.765·43-s + (−0.164 − 0.285i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.749 + 0.662i$
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: \((2,\ 1008,\ (\ :3/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.165031406\)
\(L(\frac12)\) \(\approx\) \(2.165031406\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.31 + 18.0i)T \)
good5 \( 1 + (-3.30 - 5.73i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-1.54 - 0.893i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 4.72iT - 2.19e3T^{2} \)
17 \( 1 + (23.8 - 41.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-40.1 + 23.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-30.2 + 17.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 48.3iT - 2.43e4T^{2} \)
31 \( 1 + (-107. - 62.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (137. + 238. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 37.3T + 6.89e4T^{2} \)
43 \( 1 - 215.T + 7.95e4T^{2} \)
47 \( 1 + (53.1 + 91.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-233. - 134. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-149. + 259. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (292. - 168. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-188. + 326. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 816. iT - 3.57e5T^{2} \)
73 \( 1 + (596. + 344. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-307. - 531. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.32e3T + 5.71e5T^{2} \)
89 \( 1 + (-480. - 832. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 449. iT - 9.12e5T^{2} \)
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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

−9.553142734711825449128751799450, −8.671406727828233705281150529657, −7.70834653790136932811239562327, −6.94285040724381762764203626287, −6.24607481395541027859195721036, −5.06760654087212339222307745150, −4.14795139093372696610794062092, −3.14565720952116872067096815711, −1.92463111002306514720998933163, −0.64047097418829949895897269665, 1.01032954475689477198995200077, 2.20136515423552836861003727037, 3.23718844161184543437415443562, 4.62036722244861305235213530230, 5.33487524963745899512060831374, 6.12356984870777461274567850786, 7.18562729627760183842341496849, 8.164585459465417571934872010295, 8.920030485228128879616189100273, 9.508104386720309702373525896056

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