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Properties

Label 2-57e2-19.18-c0-0-0
Degree $2$
Conductor $3249$
Sign $0.917 - 0.397i$
Analytic cond. $1.62146$
Root an. cond. $1.27336$
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  + 4-s + 7-s + 1.73i·13-s + 16-s − 25-s + 28-s + 1.73i·31-s − 1.73i·37-s − 43-s + 1.73i·52-s + 61-s + 64-s − 1.73i·67-s − 73-s − 1.73i·79-s + ⋯
L(s)  = 1  + 4-s + 7-s + 1.73i·13-s + 16-s − 25-s + 28-s + 1.73i·31-s − 1.73i·37-s − 43-s + 1.73i·52-s + 61-s + 64-s − 1.73i·67-s − 73-s − 1.73i·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3249\)    =    \(3^{2} \cdot 19^{2}\)
Sign: $0.917 - 0.397i$
Analytic conductor: \(1.62146\)
Root analytic conductor: \(1.27336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3249} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3249,\ (\ :0),\ 0.917 - 0.397i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.794523058\)
\(L(\frac12)\) \(\approx\) \(1.794523058\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.812151473651458743572696761973, −8.066762217316811310599025086328, −7.29228633293852951038404810286, −6.76285230170488720401171019306, −5.95235361466803400733115531047, −5.06936897426915465335118413157, −4.24226939882013055789386143829, −3.30135493110515896945165933230, −2.03366202953463195198326914369, −1.63304827113624410040614517528, 1.18373433663797589396647250814, 2.24075731069851620029547630904, 3.05578334199597412486305655856, 4.04602307569982952741835905998, 5.17487453540568805437357798884, 5.71195110835093055165030796696, 6.52310175908521593121712440784, 7.46052676690528089544895530621, 8.030632088042942430648472805571, 8.393072788487918333246616729286

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