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Properties

Label 2-1008-21.17-c1-0-4
Degree $2$
Conductor $1008$
Sign $0.576 - 0.817i$
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  + (1.01 + 1.75i)5-s + (−0.561 − 2.58i)7-s + (3.14 + 1.81i)11-s + 4.74i·13-s + (−1.71 + 2.97i)17-s + (−4.34 + 2.50i)19-s + (6.02 − 3.47i)23-s + (0.456 − 0.790i)25-s − 2.03i·29-s + (0.266 + 0.153i)31-s + (3.95 − 3.59i)35-s + (5.84 + 10.1i)37-s + 8.77·41-s − 3.21·43-s + (0.192 + 0.332i)47-s + ⋯
L(s)  = 1  + (0.452 + 0.783i)5-s + (−0.212 − 0.977i)7-s + (0.946 + 0.546i)11-s + 1.31i·13-s + (−0.417 + 0.722i)17-s + (−0.997 + 0.575i)19-s + (1.25 − 0.724i)23-s + (0.0912 − 0.158i)25-s − 0.378i·29-s + (0.0478 + 0.0276i)31-s + (0.669 − 0.608i)35-s + (0.961 + 1.66i)37-s + 1.37·41-s − 0.489·43-s + (0.0280 + 0.0485i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.576 - 0.817i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.576 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.686297156\)
\(L(\frac12)\) \(\approx\) \(1.686297156\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (0.561 + 2.58i)T \)
good5 \( 1 + (-1.01 - 1.75i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.14 - 1.81i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.74iT - 13T^{2} \)
17 \( 1 + (1.71 - 2.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.34 - 2.50i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.02 + 3.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.03iT - 29T^{2} \)
31 \( 1 + (-0.266 - 0.153i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.84 - 10.1i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.77T + 41T^{2} \)
43 \( 1 + 3.21T + 43T^{2} \)
47 \( 1 + (-0.192 - 0.332i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.53 - 3.19i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.25 - 7.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.05 + 0.610i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0419 + 0.0727i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.41iT - 71T^{2} \)
73 \( 1 + (9.11 + 5.26i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.821 - 1.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + (-4.30 - 7.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.7iT - 97T^{2} \)
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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

−10.16477955467365968523446275227, −9.368079889012475927637329916738, −8.529114307517564473211631470266, −7.34490163662402897333972878575, −6.52702431094836981586152707042, −6.32516731208521877188375160225, −4.49820685642732067979197159581, −4.05031266454377134863092626802, −2.65778821675942004656251842851, −1.45250412960641879429749869245, 0.847420042959029426363114093876, 2.34892887435399432528311611417, 3.40816899546927112487420391134, 4.77480692217785775836500308622, 5.52872390643248286140946053268, 6.24310356997812861325326588128, 7.32060218793291630812294877167, 8.454037572518982713768600541499, 9.127403033529045456127326958266, 9.439254626949447175077803152222

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