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Properties

Label 2-1001-7.2-c1-0-73
Degree $2$
Conductor $1001$
Sign $-0.991 - 0.126i$
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  + (0.949 − 1.64i)2-s + (−0.432 − 0.749i)3-s + (−0.804 − 1.39i)4-s + (0.432 − 0.749i)5-s − 1.64·6-s + (−2 − 1.73i)7-s + 0.743·8-s + (1.12 − 1.95i)9-s + (−0.821 − 1.42i)10-s + (0.5 + 0.866i)11-s + (−0.695 + 1.20i)12-s − 13-s + (−4.74 + 1.64i)14-s − 0.748·15-s + (2.31 − 4.00i)16-s + (−1.12 − 1.95i)17-s + ⋯
L(s)  = 1  + (0.671 − 1.16i)2-s + (−0.249 − 0.432i)3-s + (−0.402 − 0.696i)4-s + (0.193 − 0.335i)5-s − 0.670·6-s + (−0.755 − 0.654i)7-s + 0.262·8-s + (0.375 − 0.650i)9-s + (−0.259 − 0.450i)10-s + (0.150 + 0.261i)11-s + (−0.200 + 0.347i)12-s − 0.277·13-s + (−1.26 + 0.439i)14-s − 0.193·15-s + (0.578 − 1.00i)16-s + (−0.273 − 0.473i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.991 - 0.126i$
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: \((2,\ 1001,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.919136072\)
\(L(\frac12)\) \(\approx\) \(1.919136072\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2 + 1.73i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (-0.949 + 1.64i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.432 + 0.749i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.432 + 0.749i)T + (-2.5 - 4.33i)T^{2} \)
17 \( 1 + (1.12 + 1.95i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.70 + 2.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.64 - 2.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.74T + 29T^{2} \)
31 \( 1 + (4.16 + 7.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.44 - 7.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.15T + 41T^{2} \)
43 \( 1 - 8.92T + 43T^{2} \)
47 \( 1 + (-1.75 + 3.03i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.03 + 3.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.121 - 0.210i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.78 - 3.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.419 - 0.726i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.32T + 71T^{2} \)
73 \( 1 + (0.524 + 0.907i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.81 + 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + (-3.06 + 5.30i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.88T + 97T^{2} \)
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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

−9.600103708939316612441083251726, −9.254223821353685186748080002184, −7.52384933179774605970037729772, −7.09580623148615069804378191383, −5.97185892715624592841665526506, −4.90519040681326674874642045883, −3.96809401874888046707382831609, −3.19015284366301158297670803538, −1.90616597396439900450387932973, −0.72924863423106269620514761795, 2.09376378478304832199895717832, 3.54673975595993602659377974279, 4.49122986759935962502623877611, 5.47909219502775931081006283060, 5.99079743029728829713534417989, 6.85799476434800617907116496995, 7.61802218889173934380534837106, 8.604704725892159306416851872313, 9.538778936723197348615657643735, 10.50401756447477906509797767227

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