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Properties

Label 2-1001-7.2-c1-0-51
Degree $2$
Conductor $1001$
Sign $0.885 - 0.465i$
Analytic cond. $7.99302$
Root an. cond. $2.82719$
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  + (−1.27 + 2.21i)2-s + (1.33 + 2.30i)3-s + (−2.27 − 3.94i)4-s + (1.23 − 2.14i)5-s − 6.81·6-s + (−2.63 − 0.272i)7-s + 6.52·8-s + (−2.04 + 3.53i)9-s + (3.16 + 5.48i)10-s + (−0.5 − 0.866i)11-s + (6.05 − 10.4i)12-s − 13-s + (3.97 − 5.48i)14-s + 6.58·15-s + (−3.80 + 6.58i)16-s + (−3.45 − 5.98i)17-s + ⋯
L(s)  = 1  + (−0.904 + 1.56i)2-s + (0.768 + 1.33i)3-s + (−1.13 − 1.97i)4-s + (0.553 − 0.958i)5-s − 2.78·6-s + (−0.994 − 0.102i)7-s + 2.30·8-s + (−0.680 + 1.17i)9-s + (1.00 + 1.73i)10-s + (−0.150 − 0.261i)11-s + (1.74 − 3.02i)12-s − 0.277·13-s + (1.06 − 1.46i)14-s + 1.70·15-s + (−0.950 + 1.64i)16-s + (−0.837 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.885 - 0.465i$
Analytic conductor: \(7.99302\)
Root analytic conductor: \(2.82719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (716, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :1/2),\ 0.885 - 0.465i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7539285280\)
\(L(\frac12)\) \(\approx\) \(0.7539285280\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.63 + 0.272i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (1.27 - 2.21i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.33 - 2.30i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.23 + 2.14i)T + (-2.5 - 4.33i)T^{2} \)
17 \( 1 + (3.45 + 5.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.65 + 2.85i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.24 + 3.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 + (4.76 + 8.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.300 - 0.520i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.83T + 41T^{2} \)
43 \( 1 - 6.51T + 43T^{2} \)
47 \( 1 + (-2.52 + 4.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.50 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.30 - 2.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.38 + 2.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.13 + 14.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.66T + 71T^{2} \)
73 \( 1 + (3.34 + 5.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.22 - 14.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.26T + 83T^{2} \)
89 \( 1 + (6.32 - 10.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.67T + 97T^{2} \)
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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

−9.463027440196656213252427569186, −9.162348388512809092167725854797, −8.790464746353586946712998062219, −7.61596375467702694446759699685, −6.82562729079116965430331814978, −5.68757081065626782334441884871, −5.04463260009324612025467527202, −4.19467798359504615834515649330, −2.69428983977805070264981218517, −0.42293197346387020995232401970, 1.48990356890412675454154510532, 2.25183005379099320296344900403, 3.02856835317846555119033147498, 3.78038561370729352659556020432, 5.91199500081428290447022046926, 6.95051908155216607076500025522, 7.49066637878677364275234614388, 8.579011158748225997858085073185, 9.099942535620197280125640259430, 10.05794656889373308919437303792

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