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Properties

Label 2-999-37.36-c1-0-33
Degree $2$
Conductor $999$
Sign $0.904 + 0.427i$
Analytic cond. $7.97705$
Root an. cond. $2.82436$
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  + 2·4-s + 7-s − 5.19i·13-s + 4·16-s − 5.19i·19-s + 5·25-s + 2·28-s + 10.3i·31-s + (5.5 + 2.59i)37-s − 10.3i·43-s − 6·49-s − 10.3i·52-s + 15.5i·61-s + 8·64-s − 5·67-s + ⋯
L(s)  = 1  + 4-s + 0.377·7-s − 1.44i·13-s + 16-s − 1.19i·19-s + 25-s + 0.377·28-s + 1.86i·31-s + (0.904 + 0.427i)37-s − 1.58i·43-s − 0.857·49-s − 1.44i·52-s + 1.99i·61-s + 64-s − 0.610·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 999 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 999 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(999\)    =    \(3^{3} \cdot 37\)
Sign: $0.904 + 0.427i$
Analytic conductor: \(7.97705\)
Root analytic conductor: \(2.82436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{999} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 999,\ (\ :1/2),\ 0.904 + 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09681 - 0.470325i\)
\(L(\frac12)\) \(\approx\) \(2.09681 - 0.470325i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 + (-5.5 - 2.59i)T \)
good2 \( 1 - 2T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 5.19iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 15.5iT - 61T^{2} \)
67 \( 1 + 5T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + 5.19iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−10.25113730838457757421992632202, −8.970708775727529334572320771536, −8.191005004300663282348888984927, −7.32057804120496983062513471907, −6.65713911762105507784367158492, −5.59969311687957729625074763979, −4.82854933191248840480641419749, −3.31258171165782545655249386174, −2.55854356271730059640694896368, −1.10707738219417294326604830125, 1.49074209117281793383131546920, 2.45181260533058921939057205942, 3.73130298085626340887296770287, 4.74708859495628763244906057756, 5.99968177963190436725065673454, 6.55537936853374091652678039759, 7.56377311378779453474788841346, 8.168466974740101532234017702387, 9.334563920871697063557719945381, 10.01165935286159240033703753940

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