Properties

Label 2-1008-9.4-c1-0-9
Degree $2$
Conductor $1008$
Sign $-0.689 - 0.724i$
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  + (1.68 + 0.396i)3-s + (−2.18 + 3.78i)5-s + (0.5 + 0.866i)7-s + (2.68 + 1.33i)9-s + (−0.686 − 1.18i)11-s + (−1 + 1.73i)13-s + (−5.18 + 5.51i)15-s + 1.37·17-s − 5·19-s + (0.5 + 1.65i)21-s + (−0.813 + 1.40i)23-s + (−7.05 − 12.2i)25-s + (4 + 3.31i)27-s + (4.37 + 7.57i)29-s + (1 − 1.73i)31-s + ⋯
L(s)  = 1  + (0.973 + 0.228i)3-s + (−0.977 + 1.69i)5-s + (0.188 + 0.327i)7-s + (0.895 + 0.445i)9-s + (−0.206 − 0.358i)11-s + (−0.277 + 0.480i)13-s + (−1.33 + 1.42i)15-s + 0.332·17-s − 1.14·19-s + (0.109 + 0.361i)21-s + (−0.169 + 0.293i)23-s + (−1.41 − 2.44i)25-s + (0.769 + 0.638i)27-s + (0.811 + 1.40i)29-s + (0.179 − 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.689 - 0.724i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.689 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.574375349\)
\(L(\frac12)\) \(\approx\) \(1.574375349\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.396i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (2.18 - 3.78i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.686 + 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.37T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (0.813 - 1.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.37 - 7.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (2.31 - 4.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.05 + 7.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.74T + 53T^{2} \)
59 \( 1 + (5.05 - 8.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.55 + 2.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.05 - 1.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + (2.55 + 4.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.74 - 15.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + (-4.05 - 7.02i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.44880636716349393203929603115, −9.445688501186896378351527825447, −8.407216166109595873057049165265, −7.87297531576003626549815038477, −7.01332856906342144192493938645, −6.35131370279724072275923104060, −4.77792939462161263347259449504, −3.76144549455249823500204146326, −3.06037505566931324819915046008, −2.15935449074153625626220979903, 0.63088950456251519190276872361, 1.93704145215016857769358072235, 3.41545300107938504884513458680, 4.41351878975251815500381550245, 4.86468183168733485620411339317, 6.35319007362709404866567951414, 7.63557081459159520934786405812, 8.026125840458634083406238766267, 8.619376390674661935101875195419, 9.471790669745853228869820511145

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