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Properties

Label 2-1001-7.4-c1-0-72
Degree $2$
Conductor $1001$
Sign $-0.748 + 0.663i$
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  + (0.168 + 0.291i)2-s + (0.926 − 1.60i)3-s + (0.943 − 1.63i)4-s + (−1.52 − 2.63i)5-s + 0.624·6-s + (0.675 − 2.55i)7-s + 1.30·8-s + (−0.218 − 0.378i)9-s + (0.513 − 0.889i)10-s + (−0.5 + 0.866i)11-s + (−1.74 − 3.02i)12-s − 13-s + (0.860 − 0.233i)14-s − 5.64·15-s + (−1.66 − 2.88i)16-s + (2.46 − 4.26i)17-s + ⋯
L(s)  = 1  + (0.119 + 0.206i)2-s + (0.535 − 0.926i)3-s + (0.471 − 0.816i)4-s + (−0.681 − 1.18i)5-s + 0.255·6-s + (0.255 − 0.966i)7-s + 0.463·8-s + (−0.0728 − 0.126i)9-s + (0.162 − 0.281i)10-s + (−0.150 + 0.261i)11-s + (−0.504 − 0.874i)12-s − 0.277·13-s + (0.229 − 0.0625i)14-s − 1.45·15-s + (−0.416 − 0.721i)16-s + (0.597 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.748 + 0.663i$
Analytic conductor: \(7.99302\)
Root analytic conductor: \(2.82719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :1/2),\ -0.748 + 0.663i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.109444278\)
\(L(\frac12)\) \(\approx\) \(2.109444278\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.675 + 2.55i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (-0.168 - 0.291i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.926 + 1.60i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.52 + 2.63i)T + (-2.5 + 4.33i)T^{2} \)
17 \( 1 + (-2.46 + 4.26i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.10 - 5.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.496 - 0.859i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.10T + 29T^{2} \)
31 \( 1 + (2.89 - 5.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.87 - 10.1i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.70T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + (0.701 + 1.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.95 - 8.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.79 + 10.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.02 + 3.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.16 - 8.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.43T + 71T^{2} \)
73 \( 1 + (1.50 - 2.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.89 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 + (0.532 + 0.921i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.00T + 97T^{2} \)
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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

−9.767044477193295073346888863172, −8.535778739569944428285281112211, −7.73294687957397433033413099714, −7.39437999880643070171899173122, −6.44597672324426465320545853910, −5.08804385144166971809313037930, −4.66560302040033486672206444663, −3.17649703592915893282885808413, −1.61526250036168918390497251175, −0.940935823627576071138901035020, 2.37356611146404128335702054071, 3.12245607949133339895279179344, 3.73576950816557190521687602163, 4.76863724205845306117605347632, 6.10888274185183536323413306576, 7.08672306693009842248661124820, 7.85971658495593198680210047641, 8.605490639310296090211016796312, 9.462329600243253377483136716849, 10.42861022295631839098027992605

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