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Properties

Label 2-1001-7.4-c1-0-44
Degree $2$
Conductor $1001$
Sign $-0.929 + 0.369i$
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  + (−1.23 − 2.13i)2-s + (0.565 − 0.979i)3-s + (−2.05 + 3.55i)4-s + (0.362 + 0.627i)5-s − 2.79·6-s + (1.50 − 2.17i)7-s + 5.18·8-s + (0.860 + 1.49i)9-s + (0.894 − 1.54i)10-s + (−0.5 + 0.866i)11-s + (2.31 + 4.01i)12-s − 13-s + (−6.51 − 0.536i)14-s + 0.819·15-s + (−2.30 − 3.99i)16-s + (0.319 − 0.553i)17-s + ⋯
L(s)  = 1  + (−0.873 − 1.51i)2-s + (0.326 − 0.565i)3-s + (−1.02 + 1.77i)4-s + (0.162 + 0.280i)5-s − 1.14·6-s + (0.569 − 0.822i)7-s + 1.83·8-s + (0.286 + 0.497i)9-s + (0.282 − 0.490i)10-s + (−0.150 + 0.261i)11-s + (0.669 + 1.15i)12-s − 0.277·13-s + (−1.74 − 0.143i)14-s + 0.211·15-s + (−0.576 − 0.999i)16-s + (0.0775 − 0.134i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.929 + 0.369i$
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: \((2,\ 1001,\ (\ :1/2),\ -0.929 + 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.127106362\)
\(L(\frac12)\) \(\approx\) \(1.127106362\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.50 + 2.17i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (1.23 + 2.13i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.565 + 0.979i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.362 - 0.627i)T + (-2.5 + 4.33i)T^{2} \)
17 \( 1 + (-0.319 + 0.553i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.23 - 2.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.15 + 5.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.33T + 29T^{2} \)
31 \( 1 + (-2.50 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.05 + 8.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.21T + 41T^{2} \)
43 \( 1 - 7.56T + 43T^{2} \)
47 \( 1 + (5.51 + 9.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.75 - 9.96i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.39 - 5.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.218 - 0.379i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.69 + 2.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.45T + 71T^{2} \)
73 \( 1 + (-7.74 + 13.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.51 - 4.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.93T + 83T^{2} \)
89 \( 1 + (3.27 + 5.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.14T + 97T^{2} \)
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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

−9.910703468292649688477740122070, −8.880353847158135657136216810757, −7.991047534663546615103525819497, −7.61360625900327963634273257727, −6.49993568193884414544183923601, −4.79867976406519976490072815887, −3.95671858104162246491331767466, −2.64491325470866735264815624578, −1.97531705668416493423242592165, −0.77718602421547286014685830341, 1.28434635958825278790053740886, 3.12536206301684823254859983878, 4.69664112866549053831193675977, 5.26811912256715515220016267491, 6.23245996164929565573231225022, 7.02305473807870672798013808279, 8.112567484332954481266244356657, 8.504398475684629482941045251197, 9.488339494502354026976251257776, 9.649855889990070767301201287395

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