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Properties

Label 2-1001-7.2-c1-0-66
Degree $2$
Conductor $1001$
Sign $0.247 + 0.968i$
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  + (−0.820 + 1.42i)2-s + (−0.409 − 0.709i)3-s + (−0.347 − 0.602i)4-s + (1.79 − 3.10i)5-s + 1.34·6-s + (0.873 − 2.49i)7-s − 2.14·8-s + (1.16 − 2.01i)9-s + (2.94 + 5.10i)10-s + (−0.5 − 0.866i)11-s + (−0.284 + 0.493i)12-s − 13-s + (2.83 + 3.29i)14-s − 2.94·15-s + (2.45 − 4.24i)16-s + (1.46 + 2.53i)17-s + ⋯
L(s)  = 1  + (−0.580 + 1.00i)2-s + (−0.236 − 0.409i)3-s + (−0.173 − 0.301i)4-s + (0.802 − 1.38i)5-s + 0.549·6-s + (0.330 − 0.943i)7-s − 0.757·8-s + (0.388 − 0.672i)9-s + (0.931 + 1.61i)10-s + (−0.150 − 0.261i)11-s + (−0.0822 + 0.142i)12-s − 0.277·13-s + (0.757 + 0.879i)14-s − 0.759·15-s + (0.613 − 1.06i)16-s + (0.355 + 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.247 + 0.968i$
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: \((2,\ 1001,\ (\ :1/2),\ 0.247 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.047948089\)
\(L(\frac12)\) \(\approx\) \(1.047948089\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.873 + 2.49i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (0.820 - 1.42i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.409 + 0.709i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.79 + 3.10i)T + (-2.5 - 4.33i)T^{2} \)
17 \( 1 + (-1.46 - 2.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.70 - 2.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.32 + 4.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.28T + 29T^{2} \)
31 \( 1 + (2.14 + 3.70i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.23 - 9.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 - 8.72T + 43T^{2} \)
47 \( 1 + (-0.627 + 1.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.76 + 8.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.37 - 5.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.26 + 7.38i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.01 - 5.22i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.45T + 71T^{2} \)
73 \( 1 + (0.221 + 0.383i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.87 - 11.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + (-8.60 + 14.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−9.677725452658616645774125014737, −8.657328391873114450626250120814, −8.235611619269965650500937934668, −7.26624857493140994026043986114, −6.49795435931898592307955970968, −5.74032824920254334543468229229, −4.83827307110270340706139853398, −3.65664353800810745652541309636, −1.70510219469734779788967517479, −0.58773075863165540196438359855, 1.86786834264597700347104918076, 2.46466626372322782450711186633, 3.41557506010821274870060575070, 5.07826033367090439185406443134, 5.72596608161153271280284772865, 6.79953857315831092849392529830, 7.65057626516203847980552641291, 9.051220866265297519846298173653, 9.474263824146339065928558958594, 10.36912099183105267347085822812

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