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Properties

Label 2-1001-1001.373-c0-0-1
Degree $2$
Conductor $1001$
Sign $0.313 + 0.949i$
Analytic cond. $0.499564$
Root an. cond. $0.706798$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)5-s − i·7-s + i·8-s − 9-s − 0.999i·10-s − i·11-s − i·13-s + (−0.5 − 0.866i)14-s + (0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.866 + 0.5i)18-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)5-s − i·7-s + i·8-s − 9-s − 0.999i·10-s − i·11-s − i·13-s + (−0.5 − 0.866i)14-s + (0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.866 + 0.5i)18-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.313 + 0.949i$
Analytic conductor: \(0.499564\)
Root analytic conductor: \(0.706798\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :0),\ 0.313 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.526610360\)
\(L(\frac12)\) \(\approx\) \(1.526610360\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + iT \)
11 \( 1 + iT \)
13 \( 1 + iT \)
good2 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
3 \( 1 + T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.15365088011783808223976914549, −9.176934223369464829752749269578, −8.221020915877153490147895364997, −7.83745824745947485791258355722, −6.18551139614477665015700689070, −5.45960051411026427669558590122, −4.81539121706816415016639349281, −3.53767616668249323607586833508, −3.02389630563715610678037576446, −1.27141110782795538323223307708, 2.16352295700477930062509465773, 3.08166437762041813707343018860, 4.37167283371040427452548968311, 5.32826749291012833526194246462, 5.98205471170021678339744019979, 6.69558349674855098211443344162, 7.49863778964521554174869835478, 8.853316597810072721765501382280, 9.480351560060616930975078525627, 10.24833508338329631040919583094

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