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Properties

Label 2-1008-21.5-c3-0-2
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: $\chi_{1008} (593, \cdot )$
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

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

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