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Properties

Label 2-2000-5.4-c1-0-38
Degree $2$
Conductor $2000$
Sign $i$
Analytic cond. $15.9700$
Root an. cond. $3.99625$
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  + 2.65i·3-s − 4.68i·7-s − 4.06·9-s + 0.0398·11-s + 1.02i·13-s + 4.95i·17-s − 8.21·19-s + 12.4·21-s − 4.15i·23-s − 2.82i·27-s + 0.878·29-s − 5.45·31-s + 0.106i·33-s − 9.51i·37-s − 2.72·39-s + ⋯
L(s)  = 1  + 1.53i·3-s − 1.76i·7-s − 1.35·9-s + 0.0120·11-s + 0.284i·13-s + 1.20i·17-s − 1.88·19-s + 2.71·21-s − 0.866i·23-s − 0.544i·27-s + 0.163·29-s − 0.980·31-s + 0.0184i·33-s − 1.56i·37-s − 0.436·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2000\)    =    \(2^{4} \cdot 5^{3}\)
Sign: $i$
Analytic conductor: \(15.9700\)
Root analytic conductor: \(3.99625\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2000} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2000,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5395187541\)
\(L(\frac12)\) \(\approx\) \(0.5395187541\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 2.65iT - 3T^{2} \)
7 \( 1 + 4.68iT - 7T^{2} \)
11 \( 1 - 0.0398T + 11T^{2} \)
13 \( 1 - 1.02iT - 13T^{2} \)
17 \( 1 - 4.95iT - 17T^{2} \)
19 \( 1 + 8.21T + 19T^{2} \)
23 \( 1 + 4.15iT - 23T^{2} \)
29 \( 1 - 0.878T + 29T^{2} \)
31 \( 1 + 5.45T + 31T^{2} \)
37 \( 1 + 9.51iT - 37T^{2} \)
41 \( 1 + 5.28T + 41T^{2} \)
43 \( 1 + 8.95iT - 43T^{2} \)
47 \( 1 + 9.09iT - 47T^{2} \)
53 \( 1 + 1.18iT - 53T^{2} \)
59 \( 1 - 9.34T + 59T^{2} \)
61 \( 1 - 2.57T + 61T^{2} \)
67 \( 1 - 8.46iT - 67T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 + 3.58iT - 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 9.65iT - 83T^{2} \)
89 \( 1 + 0.759T + 89T^{2} \)
97 \( 1 + 6.67iT - 97T^{2} \)
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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

−8.877255620057063873222457692905, −8.531698510063459603750308872767, −7.34496503780652203670032223887, −6.62599376956240903543061702762, −5.61752178230187627171678340419, −4.53475886152973860700572543162, −4.03321569657027079837788671128, −3.60065433022028093036097326736, −1.98739612165948599183617264771, −0.18292451501830026527207850286, 1.49981313949231770174045027302, 2.37293861560177416385898975906, 3.04536739447055059082456632775, 4.71011740711591060636473436712, 5.63876483926603499098952249140, 6.25327251063519339630814793884, 6.92960458146285443147565291343, 7.85460087130612998046807783404, 8.466479056785043254385972407346, 9.075383425569638291669977599405

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