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Properties

Label 2-1008-63.25-c1-0-24
Degree $2$
Conductor $1008$
Sign $0.954 + 0.297i$
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 − 0.272i)3-s + (1.59 − 2.75i)5-s + (2.56 − 0.658i)7-s + (2.85 + 0.931i)9-s + (1.59 + 2.75i)11-s + (2.85 + 4.93i)13-s + (−3.47 + 4.28i)15-s + (−0.760 + 1.31i)17-s + (0.641 + 1.11i)19-s + (−4.56 + 0.429i)21-s + (1.11 − 1.93i)23-s + (−2.56 − 4.43i)25-s + (−4.62 − 2.36i)27-s + (−3.54 + 6.13i)29-s + 9.42·31-s + ⋯
L(s)  = 1  + (−0.987 − 0.157i)3-s + (0.711 − 1.23i)5-s + (0.968 − 0.249i)7-s + (0.950 + 0.310i)9-s + (0.479 + 0.830i)11-s + (0.790 + 1.36i)13-s + (−0.896 + 1.10i)15-s + (−0.184 + 0.319i)17-s + (0.147 + 0.254i)19-s + (−0.995 + 0.0937i)21-s + (0.233 − 0.404i)23-s + (−0.512 − 0.887i)25-s + (−0.890 − 0.455i)27-s + (−0.657 + 1.13i)29-s + 1.69·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.954 + 0.297i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.954 + 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.608627606\)
\(L(\frac12)\) \(\approx\) \(1.608627606\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.272i)T \)
7 \( 1 + (-2.56 + 0.658i)T \)
good5 \( 1 + (-1.59 + 2.75i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.59 - 2.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.85 - 4.93i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.760 - 1.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.641 - 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.11 + 1.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.54 - 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.42T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.80 + 4.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.41 - 5.91i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.82T + 47T^{2} \)
53 \( 1 + (-1.02 + 1.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.12T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + 8.69T + 71T^{2} \)
73 \( 1 + (2.48 - 4.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 4.13T + 79T^{2} \)
83 \( 1 + (-4.03 + 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.112 - 0.195i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.42 + 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.957173461059692076570079714244, −9.075801944981360473633126512958, −8.427903964541388156236433908948, −7.23748672382274982862368867013, −6.44933413144962526636686750370, −5.53411158092453626380177746255, −4.65289890291381468538172867432, −4.23275347176796323784113961632, −1.77809723784049884372201133048, −1.28709742138767331941989284185, 1.07603075413844062980935881342, 2.58701381403398510692258853420, 3.72145895952480140282974803885, 5.01478102127000862832057924351, 5.87923822569158399210393013669, 6.31151315608636840075523768685, 7.36515244883759592909830457533, 8.268133404281656735383757452160, 9.361282922107553191436848500822, 10.37376314961998769210972610282

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