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Properties

Label 2-1008-9.4-c1-0-20
Degree $2$
Conductor $1008$
Sign $1$
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  + (1.70 − 0.300i)3-s + (−0.266 + 0.460i)5-s + (0.5 + 0.866i)7-s + (2.81 − 1.02i)9-s + (−1.11 − 1.92i)11-s + (−1.03 + 1.78i)13-s + (−0.315 + 0.866i)15-s + 0.815·17-s + 7.94·19-s + (1.11 + 1.32i)21-s + (3.40 − 5.88i)23-s + (2.35 + 4.08i)25-s + (4.49 − 2.59i)27-s + (3.73 + 6.47i)29-s + (1.14 − 1.98i)31-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)3-s + (−0.118 + 0.206i)5-s + (0.188 + 0.327i)7-s + (0.939 − 0.342i)9-s + (−0.335 − 0.581i)11-s + (−0.286 + 0.495i)13-s + (−0.0813 + 0.223i)15-s + 0.197·17-s + 1.82·19-s + (0.242 + 0.289i)21-s + (0.709 − 1.22i)23-s + (0.471 + 0.816i)25-s + (0.866 − 0.499i)27-s + (0.694 + 1.20i)29-s + (0.205 − 0.356i)31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.364283167\)
\(L(\frac12)\) \(\approx\) \(2.364283167\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.300i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.266 - 0.460i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.11 + 1.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.03 - 1.78i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.815T + 17T^{2} \)
19 \( 1 - 7.94T + 19T^{2} \)
23 \( 1 + (-3.40 + 5.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.73 - 6.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.14 + 1.98i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + (-1.29 + 2.24i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.145 - 0.251i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.213 - 0.368i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.41T + 53T^{2} \)
59 \( 1 + (-1.71 + 2.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.23 - 9.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.10 - 3.64i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.0T + 71T^{2} \)
73 \( 1 + 9.09T + 73T^{2} \)
79 \( 1 + (-3.73 - 6.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.76 + 15.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.09T + 89T^{2} \)
97 \( 1 + (5.94 + 10.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.875760445882749406633584371407, −8.888723471429106679502576412703, −8.533641047654899730292042971278, −7.34876742471896675095022235190, −6.96610945093200811047032444564, −5.58875355041247794476006011019, −4.66042793191578956307680875446, −3.35651725220645905368586818790, −2.74343445856283867004948612253, −1.31121132732396037901066303278, 1.27412356874032592450559692955, 2.68634789574734476642563894071, 3.54631601756940740412196301458, 4.67671922338897565095833053415, 5.39030695747320763867833658874, 6.93339027216556866987405652582, 7.58760242094766658839718823729, 8.215325529316221595686892012850, 9.190565624833040630786779226520, 9.918038105182505552716353097595

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