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Properties

Label 2-1003-17.16-c1-0-0
Degree $2$
Conductor $1003$
Sign $0.516 + 0.856i$
Analytic cond. $8.00899$
Root an. cond. $2.83001$
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  − 2.79·2-s + 3.34i·3-s + 5.80·4-s + 0.976i·5-s − 9.36i·6-s − 0.327i·7-s − 10.6·8-s − 8.22·9-s − 2.72i·10-s + 3.54i·11-s + 19.4i·12-s − 4.81·13-s + 0.916i·14-s − 3.27·15-s + 18.1·16-s + (−2.13 − 3.53i)17-s + ⋯
L(s)  = 1  − 1.97·2-s + 1.93i·3-s + 2.90·4-s + 0.436i·5-s − 3.82i·6-s − 0.123i·7-s − 3.76·8-s − 2.74·9-s − 0.863i·10-s + 1.06i·11-s + 5.61i·12-s − 1.33·13-s + 0.244i·14-s − 0.844·15-s + 4.52·16-s + (−0.516 − 0.856i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1003\)    =    \(17 \cdot 59\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(8.00899\)
Root analytic conductor: \(2.83001\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1003} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1003,\ (\ :1/2),\ 0.516 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06089309696\)
\(L(\frac12)\) \(\approx\) \(0.06089309696\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.13 + 3.53i)T \)
59 \( 1 + T \)
good2 \( 1 + 2.79T + 2T^{2} \)
3 \( 1 - 3.34iT - 3T^{2} \)
5 \( 1 - 0.976iT - 5T^{2} \)
7 \( 1 + 0.327iT - 7T^{2} \)
11 \( 1 - 3.54iT - 11T^{2} \)
13 \( 1 + 4.81T + 13T^{2} \)
19 \( 1 + 1.29T + 19T^{2} \)
23 \( 1 - 1.13iT - 23T^{2} \)
29 \( 1 - 8.43iT - 29T^{2} \)
31 \( 1 + 4.61iT - 31T^{2} \)
37 \( 1 + 5.97iT - 37T^{2} \)
41 \( 1 - 3.21iT - 41T^{2} \)
43 \( 1 + 4.02T + 43T^{2} \)
47 \( 1 - 6.04T + 47T^{2} \)
53 \( 1 + 5.56T + 53T^{2} \)
61 \( 1 - 7.69iT - 61T^{2} \)
67 \( 1 + 1.62T + 67T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + 4.11iT - 73T^{2} \)
79 \( 1 + 3.10iT - 79T^{2} \)
83 \( 1 + 9.72T + 83T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−10.50483491230099719421505231370, −9.627518938193745713696139208485, −9.376548033067027347556321562865, −8.607308663768216377336676075678, −7.48376453645104532206340231022, −6.86233410110376765093489624556, −5.57358580647486247798994259733, −4.53994058744114518333178615194, −3.10033146787918626163262110727, −2.33810706759624180626545997540, 0.05738964779574855038760595803, 1.08806596714442667154365980855, 2.14777431481258011430113457337, 2.89367172492256283193081605681, 5.61054271734806611696798388606, 6.39614495416979199726990492256, 7.00323480126203994334193173438, 7.79911265199473420296293156018, 8.501297664618165175634599046713, 8.769240987276840180191461060889

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