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Properties

Label 2-1008-63.4-c1-0-0
Degree $2$
Conductor $1008$
Sign $-0.778 + 0.627i$
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  + (1.09 + 1.34i)3-s − 3.18·5-s + (−0.710 + 2.54i)7-s + (−0.619 + 2.93i)9-s − 3.18·11-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 + (−4.20 + 1.82i)21-s − 2.23·23-s + 5.12·25-s + (−4.62 + 2.36i)27-s + (−3.54 − 6.13i)29-s + (−4.71 − 8.15i)31-s + ⋯
L(s)  = 1  + (0.629 + 0.776i)3-s − 1.42·5-s + (−0.268 + 0.963i)7-s + (−0.206 + 0.978i)9-s − 0.959·11-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 + (−0.917 + 0.398i)21-s − 0.466·23-s + 1.02·25-s + (−0.890 + 0.455i)27-s + (−0.657 − 1.13i)29-s + (−0.846 − 1.46i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.778 + 0.627i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.778 + 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1923301533\)
\(L(\frac12)\) \(\approx\) \(0.1923301533\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 - 1.34i)T \)
7 \( 1 + (0.710 - 2.54i)T \)
good5 \( 1 + 3.18T + 5T^{2} \)
11 \( 1 + 3.18T + 11T^{2} \)
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 \( 1 + 2.23T + 23T^{2} \)
29 \( 1 + (3.54 + 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.71 + 8.15i)T + (-15.5 + 26.8i)T^{2} \)
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 \( 1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.02 + 1.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.562 + 0.974i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.56 - 2.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.48 - 9.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.69T + 71T^{2} \)
73 \( 1 + (2.48 - 4.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.06 - 3.58i)T + (-39.5 - 68.4i)T^{2} \)
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

−10.42066943919232051246483867101, −9.650248812943599633622930048733, −8.603446330864757186610601066805, −8.054584434540786729480482717287, −7.62062361260721963471603024171, −5.97412594588658637327676817615, −5.24055674252385429145576485966, −4.05919675343856263197302591918, −3.38129474673033735735346910015, −2.42806156373141011889408274115, 0.07882457842957496971565554576, 1.63760455182203887238887535590, 3.24469110894255092947684377146, 3.80427775696691083603529646305, 4.86371967576363537346494634352, 6.43492667765884016096970498390, 7.19819250737877158700857456926, 7.61029259814213742664079669601, 8.546365467497466462437341153377, 9.156634918457299439747391616287

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