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Properties

Label 2-2001-2001.68-c0-0-4
Degree $2$
Conductor $2001$
Sign $-0.521 + 0.853i$
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  + (−0.0397 − 0.0633i)2-s + (−0.680 − 0.733i)3-s + (0.431 − 0.895i)4-s + (−0.0193 + 0.0722i)6-s + (−0.148 + 0.0166i)8-s + (−0.0747 + 0.997i)9-s + (−0.950 + 0.293i)12-s + (1.14 − 0.914i)13-s + (−0.613 − 0.768i)16-s + (0.0661 − 0.0349i)18-s + (0.974 − 0.222i)23-s + (0.113 + 0.0972i)24-s + (−0.900 − 0.433i)25-s + (−0.103 − 0.0362i)26-s + (0.781 − 0.623i)27-s + ⋯
L(s)  = 1  + (−0.0397 − 0.0633i)2-s + (−0.680 − 0.733i)3-s + (0.431 − 0.895i)4-s + (−0.0193 + 0.0722i)6-s + (−0.148 + 0.0166i)8-s + (−0.0747 + 0.997i)9-s + (−0.950 + 0.293i)12-s + (1.14 − 0.914i)13-s + (−0.613 − 0.768i)16-s + (0.0661 − 0.0349i)18-s + (0.974 − 0.222i)23-s + (0.113 + 0.0972i)24-s + (−0.900 − 0.433i)25-s + (−0.103 − 0.0362i)26-s + (0.781 − 0.623i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.521 + 0.853i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.521 + 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9599641818\)
\(L(\frac12)\) \(\approx\) \(0.9599641818\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.680 + 0.733i)T \)
23 \( 1 + (-0.974 + 0.222i)T \)
29 \( 1 + (0.563 + 0.826i)T \)
good2 \( 1 + (0.0397 + 0.0633i)T + (-0.433 + 0.900i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
7 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.974 - 0.222i)T^{2} \)
13 \( 1 + (-1.14 + 0.914i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.781 - 0.623i)T^{2} \)
31 \( 1 + (-0.438 + 0.275i)T + (0.433 - 0.900i)T^{2} \)
37 \( 1 + (0.974 - 0.222i)T^{2} \)
41 \( 1 + (1.29 + 1.29i)T + iT^{2} \)
43 \( 1 + (0.433 + 0.900i)T^{2} \)
47 \( 1 + (0.220 - 1.95i)T + (-0.974 - 0.222i)T^{2} \)
53 \( 1 + (-0.900 - 0.433i)T^{2} \)
59 \( 1 - 1.80iT - T^{2} \)
61 \( 1 + (-0.781 - 0.623i)T^{2} \)
67 \( 1 + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (-0.367 - 0.460i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (1.10 + 0.694i)T + (0.433 + 0.900i)T^{2} \)
79 \( 1 + (-0.974 + 0.222i)T^{2} \)
83 \( 1 + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.433 - 0.900i)T^{2} \)
97 \( 1 + (-0.781 + 0.623i)T^{2} \)
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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

−9.077415422008836950907986909260, −8.191070698969821906695919955951, −7.39445309784942313218576711969, −6.59366459017816815302472143387, −5.86511813247032563032661218240, −5.45823551649250583846465754987, −4.32996163613712917388550511070, −2.92175087505596034133372270271, −1.84226344630039793756909745616, −0.795278527984430281570936098745, 1.65883462935530359935140044456, 3.21470167729418960411468033771, 3.76554940781497190449981415617, 4.69260072984426803421111163293, 5.65230083687985259693608158159, 6.57999908663930433437684073309, 7.02467888426352839855863615005, 8.186299915367010551857199622117, 8.857879241607461861520601769642, 9.519926201524304676711243739871

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