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Properties

Label 2-2001-2001.965-c0-0-5
Degree $2$
Conductor $2001$
Sign $-0.253 - 0.967i$
Analytic cond. $0.998629$
Root an. cond. $0.999314$
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  + (0.623 − 1.78i)2-s + (0.433 − 0.900i)3-s + (−2.00 − 1.59i)4-s + (−1.33 − 1.33i)6-s + (−2.49 + 1.57i)8-s + (−0.623 − 0.781i)9-s + (−2.30 + 1.11i)12-s + (−1.75 + 0.400i)13-s + (0.669 + 2.93i)16-s + (−1.78 + 0.623i)18-s + (0.433 − 0.900i)23-s + (0.330 + 2.93i)24-s + (0.623 − 0.781i)25-s + (−0.380 + 3.38i)26-s + (−0.974 + 0.222i)27-s + ⋯
L(s)  = 1  + (0.623 − 1.78i)2-s + (0.433 − 0.900i)3-s + (−2.00 − 1.59i)4-s + (−1.33 − 1.33i)6-s + (−2.49 + 1.57i)8-s + (−0.623 − 0.781i)9-s + (−2.30 + 1.11i)12-s + (−1.75 + 0.400i)13-s + (0.669 + 2.93i)16-s + (−1.78 + 0.623i)18-s + (0.433 − 0.900i)23-s + (0.330 + 2.93i)24-s + (0.623 − 0.781i)25-s + (−0.380 + 3.38i)26-s + (−0.974 + 0.222i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.253 - 0.967i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (965, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.253 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.224492877\)
\(L(\frac12)\) \(\approx\) \(1.224492877\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.433 + 0.900i)T \)
23 \( 1 + (-0.433 + 0.900i)T \)
29 \( 1 + (0.781 + 0.623i)T \)
good2 \( 1 + (-0.623 + 1.78i)T + (-0.781 - 0.623i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
7 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (-0.433 - 0.900i)T^{2} \)
13 \( 1 + (1.75 - 0.400i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-0.974 + 0.222i)T^{2} \)
31 \( 1 + (-1.33 - 0.467i)T + (0.781 + 0.623i)T^{2} \)
37 \( 1 + (0.433 - 0.900i)T^{2} \)
41 \( 1 + (0.752 + 0.752i)T + iT^{2} \)
43 \( 1 + (0.781 - 0.623i)T^{2} \)
47 \( 1 + (-1.05 + 1.68i)T + (-0.433 - 0.900i)T^{2} \)
53 \( 1 + (0.623 - 0.781i)T^{2} \)
59 \( 1 + 1.24iT - T^{2} \)
61 \( 1 + (0.974 + 0.222i)T^{2} \)
67 \( 1 + (-0.900 - 0.433i)T^{2} \)
71 \( 1 + (-0.193 - 0.846i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (-1.87 + 0.656i)T + (0.781 - 0.623i)T^{2} \)
79 \( 1 + (-0.433 + 0.900i)T^{2} \)
83 \( 1 + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.781 + 0.623i)T^{2} \)
97 \( 1 + (0.974 - 0.222i)T^{2} \)
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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

−9.008378920509109949471986378988, −8.304923947830623527281473904371, −7.19893971153842422921942951791, −6.34001055971070927083593078725, −5.19509665581139674737742817206, −4.53135710819732794873974862876, −3.48260005380094972863155331358, −2.51372762243530060108187544018, −2.10379207565987332566114184292, −0.66751816489813464279854681340, 2.75443183476180981967800331239, 3.59287744867740443558147174250, 4.61688692628966041352061418788, 5.07847042368699786486483923993, 5.74870096407138194464285945737, 6.84615865309500115865712514675, 7.56642689952843953414908251196, 8.052916756218446822845185075708, 9.043537416749684112895792750937, 9.481658734099226822035723487948

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