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Properties

Label 2-1008-63.4-c1-0-36
Degree $2$
Conductor $1008$
Sign $0.823 + 0.566i$
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  + (1.5 − 0.866i)3-s + 5-s + (2.5 − 0.866i)7-s + (1.5 − 2.59i)9-s + 3·11-s + (−0.5 + 0.866i)13-s + (1.5 − 0.866i)15-s + (−1.5 + 2.59i)17-s + (2.5 + 4.33i)19-s + (3 − 3.46i)21-s − 23-s − 4·25-s − 5.19i·27-s + (−4.5 − 7.79i)29-s + (2 + 3.46i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + 0.447·5-s + (0.944 − 0.327i)7-s + (0.5 − 0.866i)9-s + 0.904·11-s + (−0.138 + 0.240i)13-s + (0.387 − 0.223i)15-s + (−0.363 + 0.630i)17-s + (0.573 + 0.993i)19-s + (0.654 − 0.755i)21-s − 0.208·23-s − 0.800·25-s − 0.999i·27-s + (−0.835 − 1.44i)29-s + (0.359 + 0.622i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.823 + 0.566i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.823 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.661830726\)
\(L(\frac12)\) \(\approx\) \(2.661830726\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 - T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.5 - 2.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.5 + 11.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.718064825195812181491392346196, −9.040151919805813430038464962107, −8.118943662606817609951073246647, −7.58504368879306099563759295735, −6.56404412218236368100298542352, −5.74309446452027759966945853436, −4.35161592462145940197429539428, −3.63520668692261658897688127217, −2.13329657963466697794068030030, −1.39928190328830203230809695440, 1.59898483877940763721174448712, 2.60927833244559101138015319216, 3.76790532616847037004619472395, 4.79243623790019665119544464351, 5.49046248812668576999836412958, 6.84199170077041713333442427253, 7.64715858611448764716170142053, 8.572924509691368573195944604944, 9.217845918084275416428514865098, 9.779234120622260103825673264826

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