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Properties

Label 2-1008-48.35-c1-0-41
Degree $2$
Conductor $1008$
Sign $0.997 - 0.0656i$
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  + (1.39 − 0.221i)2-s + (1.90 − 0.618i)4-s + (2.51 + 2.51i)5-s + 7-s + (2.52 − 1.28i)8-s + (4.06 + 2.95i)10-s + (0.984 − 0.984i)11-s + (−3.26 − 3.26i)13-s + (1.39 − 0.221i)14-s + (3.23 − 2.35i)16-s − 5.50i·17-s + (−1.21 + 1.21i)19-s + (6.33 + 3.22i)20-s + (1.15 − 1.59i)22-s + 8.62i·23-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s + (0.951 − 0.309i)4-s + (1.12 + 1.12i)5-s + 0.377·7-s + (0.891 − 0.453i)8-s + (1.28 + 0.934i)10-s + (0.296 − 0.296i)11-s + (−0.904 − 0.904i)13-s + (0.373 − 0.0591i)14-s + (0.809 − 0.587i)16-s − 1.33i·17-s + (−0.278 + 0.278i)19-s + (1.41 + 0.722i)20-s + (0.246 − 0.339i)22-s + 1.79i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.997 - 0.0656i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.997 - 0.0656i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.719050044\)
\(L(\frac12)\) \(\approx\) \(3.719050044\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.221i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (-2.51 - 2.51i)T + 5iT^{2} \)
11 \( 1 + (-0.984 + 0.984i)T - 11iT^{2} \)
13 \( 1 + (3.26 + 3.26i)T + 13iT^{2} \)
17 \( 1 + 5.50iT - 17T^{2} \)
19 \( 1 + (1.21 - 1.21i)T - 19iT^{2} \)
23 \( 1 - 8.62iT - 23T^{2} \)
29 \( 1 + (-2.04 + 2.04i)T - 29iT^{2} \)
31 \( 1 - 0.164iT - 31T^{2} \)
37 \( 1 + (8.31 - 8.31i)T - 37iT^{2} \)
41 \( 1 + 9.56T + 41T^{2} \)
43 \( 1 + (-5.01 - 5.01i)T + 43iT^{2} \)
47 \( 1 + 3.37T + 47T^{2} \)
53 \( 1 + (3.72 + 3.72i)T + 53iT^{2} \)
59 \( 1 + (-10.0 + 10.0i)T - 59iT^{2} \)
61 \( 1 + (1.07 + 1.07i)T + 61iT^{2} \)
67 \( 1 + (3.12 - 3.12i)T - 67iT^{2} \)
71 \( 1 + 7.66iT - 71T^{2} \)
73 \( 1 + 8.40iT - 73T^{2} \)
79 \( 1 - 13.8iT - 79T^{2} \)
83 \( 1 + (-7.31 - 7.31i)T + 83iT^{2} \)
89 \( 1 + 7.49T + 89T^{2} \)
97 \( 1 + 7.84T + 97T^{2} \)
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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

−9.961091377305099314254742361179, −9.677998318417848778040441068744, −8.046608500705346847633792984301, −7.11493002697716553644604593823, −6.51753859695809888806883299942, −5.49232583672639333677477751425, −5.00153928079831733982487678446, −3.42660569026859137720323457125, −2.75596800180646564975866504054, −1.68114833650911578478393855383, 1.63405225444160007546789682508, 2.34319124545539480136081028891, 4.07367714722157508293031678590, 4.74118819367766811649158276903, 5.48915387683646732781075360106, 6.39432398012647039858104850412, 7.13379973552546297808098091411, 8.456108167903460087905788954227, 8.934216415388635012803896051517, 10.12748449564081941442293751698

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