Properties

Label 2-1008-21.17-c3-0-45
Degree $2$
Conductor $1008$
Sign $-0.945 - 0.326i$
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  + (−7.54 − 13.0i)5-s + (−16.2 + 8.80i)7-s + (8.56 + 4.94i)11-s − 67.8i·13-s + (35.0 − 60.7i)17-s + (53.2 − 30.7i)19-s + (113. − 65.7i)23-s + (−51.3 + 88.8i)25-s + 158. i·29-s + (66.2 + 38.2i)31-s + (237. + 146. i)35-s + (−174. − 301. i)37-s + 138.·41-s − 539.·43-s + (−111. − 193. i)47-s + ⋯
L(s)  = 1  + (−0.674 − 1.16i)5-s + (−0.879 + 0.475i)7-s + (0.234 + 0.135i)11-s − 1.44i·13-s + (0.500 − 0.866i)17-s + (0.642 − 0.371i)19-s + (1.03 − 0.596i)23-s + (−0.410 + 0.711i)25-s + 1.01i·29-s + (0.383 + 0.221i)31-s + (1.14 + 0.707i)35-s + (−0.774 − 1.34i)37-s + 0.529·41-s − 1.91·43-s + (−0.347 − 0.601i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.945 - 0.326i$
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: \((2,\ 1008,\ (\ :3/2),\ -0.945 - 0.326i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6037087388\)
\(L(\frac12)\) \(\approx\) \(0.6037087388\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.2 - 8.80i)T \)
good5 \( 1 + (7.54 + 13.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-8.56 - 4.94i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 67.8iT - 2.19e3T^{2} \)
17 \( 1 + (-35.0 + 60.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-53.2 + 30.7i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-113. + 65.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 158. iT - 2.43e4T^{2} \)
31 \( 1 + (-66.2 - 38.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (174. + 301. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 138.T + 6.89e4T^{2} \)
43 \( 1 + 539.T + 7.95e4T^{2} \)
47 \( 1 + (111. + 193. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (459. + 265. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (271. - 470. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (116. - 67.0i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-160. + 277. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 416. iT - 3.57e5T^{2} \)
73 \( 1 + (-472. - 272. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (161. + 279. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 885.T + 5.71e5T^{2} \)
89 \( 1 + (-812. - 1.40e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 739. iT - 9.12e5T^{2} \)
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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

−9.048265669340342119033335055470, −8.428378698762853151372660259995, −7.51650788968659053379836492763, −6.64518760629514375695518419622, −5.33002547266658642550100107482, −5.00424746294670488862139422148, −3.59178592887670661451040590951, −2.84303008409876191026805514462, −1.05543795034336738480659032940, −0.17801668914702054666568827992, 1.50674713455854152383680908063, 3.08809086911956478654608533136, 3.58419783898535929880061940682, 4.60801486413078396322761899727, 6.12592145915605203062342346018, 6.69371808761845553148504428509, 7.37915179898963701530502824277, 8.249153145041323274642159268946, 9.441628030820274171255616909025, 9.978309326591241309402461292428

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