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Properties

Label 2-2997-333.221-c0-0-4
Degree $2$
Conductor $2997$
Sign $0.984 - 0.173i$
Analytic cond. $1.49569$
Root an. cond. $1.22298$
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  + (0.5 − 0.866i)4-s + (1 + 1.73i)7-s + (−0.499 − 0.866i)16-s + (0.5 + 0.866i)25-s + 1.99·28-s + 37-s + (−1.49 + 2.59i)49-s − 0.999·64-s + (1 − 1.73i)67-s − 2·73-s + 0.999·100-s + (0.999 − 1.73i)112-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (1 + 1.73i)7-s + (−0.499 − 0.866i)16-s + (0.5 + 0.866i)25-s + 1.99·28-s + 37-s + (−1.49 + 2.59i)49-s − 0.999·64-s + (1 − 1.73i)67-s − 2·73-s + 0.999·100-s + (0.999 − 1.73i)112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2997\)    =    \(3^{4} \cdot 37\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(1.49569\)
Root analytic conductor: \(1.22298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2997} (1997, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2997,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.576712689\)
\(L(\frac12)\) \(\approx\) \(1.576712689\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 - T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 2T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.045874274066077103004352182923, −8.237635100861297439862241327400, −7.47939694651670707553647792390, −6.48781489812290071713301704118, −5.79112591537766658063245373941, −5.24379064977879958144406345405, −4.55563166227863818053339537367, −3.02896300407021348014147865755, −2.22606504896827537923395871780, −1.42668485101681233818147760024, 1.12853719537402728566945256068, 2.30036691477415754919290428501, 3.39640382741699601130382273782, 4.22373444376055968439483399050, 4.69989830586883992679718244604, 5.98448843354096608842859718933, 7.00029018998915983505334823165, 7.29029928983728575949177202850, 8.141245857599823274640726811367, 8.519729913815581603725134377467

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