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Properties

Label 2-1008-252.223-c1-0-14
Degree $2$
Conductor $1008$
Sign $0.989 + 0.145i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.71 − 0.226i)3-s + (−1.72 + 0.993i)5-s + (−2.57 − 0.612i)7-s + (2.89 + 0.777i)9-s + (−2.20 − 1.27i)11-s + (−1.56 + 0.901i)13-s + (3.17 − 1.31i)15-s − 0.0160i·17-s − 2.71·19-s + (4.28 + 1.63i)21-s + (5.44 − 3.14i)23-s + (−0.527 + 0.913i)25-s + (−4.79 − 1.99i)27-s + (0.871 − 1.50i)29-s + (3.82 + 6.62i)31-s + ⋯
L(s)  = 1  + (−0.991 − 0.130i)3-s + (−0.769 + 0.444i)5-s + (−0.972 − 0.231i)7-s + (0.965 + 0.259i)9-s + (−0.663 − 0.383i)11-s + (−0.432 + 0.249i)13-s + (0.820 − 0.339i)15-s − 0.00388i·17-s − 0.621·19-s + (0.934 + 0.356i)21-s + (1.13 − 0.655i)23-s + (−0.105 + 0.182i)25-s + (−0.923 − 0.382i)27-s + (0.161 − 0.280i)29-s + (0.687 + 1.19i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.989 + 0.145i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.989 + 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6362135736\)
\(L(\frac12)\) \(\approx\) \(0.6362135736\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.226i)T \)
7 \( 1 + (2.57 + 0.612i)T \)
good5 \( 1 + (1.72 - 0.993i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.20 + 1.27i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.56 - 0.901i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.0160iT - 17T^{2} \)
19 \( 1 + 2.71T + 19T^{2} \)
23 \( 1 + (-5.44 + 3.14i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.871 + 1.50i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.82 - 6.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.20T + 37T^{2} \)
41 \( 1 + (0.455 - 0.262i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.62 + 2.67i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.99 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.61T + 53T^{2} \)
59 \( 1 + (-1.95 - 3.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.31 - 5.37i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.84 - 1.64i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.552iT - 71T^{2} \)
73 \( 1 + 9.52iT - 73T^{2} \)
79 \( 1 + (-6.04 - 3.48i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.60 + 9.71i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.31iT - 89T^{2} \)
97 \( 1 + (-2.73 - 1.58i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.38845290681048499132351854465, −9.209466253895880003231614221164, −8.137423936744247767195770511020, −7.11586081848981402697743123894, −6.73007838513807026880668595803, −5.71912288867388788116742607005, −4.71625036489591913978040572524, −3.73668971295958623465548987854, −2.60522526697502466715584970215, −0.59430548372001169700829239236, 0.66437576793932348244913380419, 2.62176222496968303415630496349, 3.94629527054029448088535834704, 4.75573559496379813381573473653, 5.65711748158567737816240370971, 6.53845420590077614271503488182, 7.39891467517266672599772729415, 8.212562540189839238251961000055, 9.427197607974734477001197743160, 9.931826647911126020132597239057

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