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Properties

Label 2-1008-63.59-c1-0-8
Degree $2$
Conductor $1008$
Sign $0.754 - 0.656i$
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  + (−0.615 − 1.61i)3-s − 3.91·5-s + (2.51 + 0.813i)7-s + (−2.24 + 1.99i)9-s − 3.69i·11-s + (−0.480 − 0.277i)13-s + (2.41 + 6.33i)15-s + (−2.91 + 5.05i)17-s + (−4.62 + 2.66i)19-s + (−0.233 − 4.57i)21-s + 2.27i·23-s + 10.3·25-s + (4.60 + 2.40i)27-s + (3.53 − 2.04i)29-s + (7.00 − 4.04i)31-s + ⋯
L(s)  = 1  + (−0.355 − 0.934i)3-s − 1.75·5-s + (0.951 + 0.307i)7-s + (−0.747 + 0.664i)9-s − 1.11i·11-s + (−0.133 − 0.0769i)13-s + (0.622 + 1.63i)15-s + (−0.707 + 1.22i)17-s + (−1.06 + 0.612i)19-s + (−0.0509 − 0.998i)21-s + 0.474i·23-s + 2.06·25-s + (0.886 + 0.461i)27-s + (0.656 − 0.379i)29-s + (1.25 − 0.726i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.754 - 0.656i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.754 - 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6982351056\)
\(L(\frac12)\) \(\approx\) \(0.6982351056\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.615 + 1.61i)T \)
7 \( 1 + (-2.51 - 0.813i)T \)
good5 \( 1 + 3.91T + 5T^{2} \)
11 \( 1 + 3.69iT - 11T^{2} \)
13 \( 1 + (0.480 + 0.277i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.91 - 5.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.62 - 2.66i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.27iT - 23T^{2} \)
29 \( 1 + (-3.53 + 2.04i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.00 + 4.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.89 - 6.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.59 - 6.22i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.754 - 1.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0415 - 0.0239i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.45 - 7.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.03 - 3.48i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.587 - 1.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.71iT - 71T^{2} \)
73 \( 1 + (3.52 + 2.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.97 - 3.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.84 - 6.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.71 - 4.69i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.9 - 8.07i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.45115481748541475417323237766, −8.623181396052506008871850593902, −8.195503681026804639497142738998, −7.890295297838325848569401592416, −6.68436318036688405687116488106, −5.95770687586993166295405414981, −4.72991045676361449881472497005, −3.90442076215913832784391505330, −2.59328360050968480676421128164, −1.08735118851998235617937731897, 0.40809595156956437599519558856, 2.61200751121310974082466339501, 3.97451127281625186790875588850, 4.56738701867477914690632043852, 4.99087820237925356966876082803, 6.73790215885871948192613098198, 7.30921351128839827950249660671, 8.356193288084634928498042766636, 8.875750626701734989531526598118, 10.03438766549421810418079830910

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