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Properties

Label 2-57e2-19.3-c0-0-2
Degree $2$
Conductor $3249$
Sign $0.944 + 0.327i$
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  + (0.173 + 0.984i)4-s + (−0.5 − 0.866i)7-s + (0.592 − 1.62i)13-s + (−0.939 + 0.342i)16-s + (0.939 + 0.342i)25-s + (0.766 − 0.642i)28-s + (1.5 − 0.866i)31-s − 1.73i·37-s + (−0.173 + 0.984i)43-s + (1.70 + 0.300i)52-s + (0.173 + 0.984i)61-s + (−0.5 − 0.866i)64-s + (−1.11 − 1.32i)67-s + (0.939 − 0.342i)73-s + (0.592 + 1.62i)79-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)4-s + (−0.5 − 0.866i)7-s + (0.592 − 1.62i)13-s + (−0.939 + 0.342i)16-s + (0.939 + 0.342i)25-s + (0.766 − 0.642i)28-s + (1.5 − 0.866i)31-s − 1.73i·37-s + (−0.173 + 0.984i)43-s + (1.70 + 0.300i)52-s + (0.173 + 0.984i)61-s + (−0.5 − 0.866i)64-s + (−1.11 − 1.32i)67-s + (0.939 − 0.342i)73-s + (0.592 + 1.62i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3249\)    =    \(3^{2} \cdot 19^{2}\)
Sign: $0.944 + 0.327i$
Analytic conductor: \(1.62146\)
Root analytic conductor: \(1.27336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3249} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3249,\ (\ :0),\ 0.944 + 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.235440933\)
\(L(\frac12)\) \(\approx\) \(1.235440933\)
\(L(1)\) 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 \( 1 + (-0.173 - 0.984i)T^{2} \)
5 \( 1 + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T^{2} \)
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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

−8.641027113855411224389508520707, −7.929399562690597163970697361600, −7.46327401811870177769543594048, −6.63359538927004450783998160230, −5.91263642089591110989436297253, −4.83822763645264864411018635362, −3.90836901580677200872982013769, −3.29188700731817351417603158541, −2.51411093538820049587823995574, −0.841413954928447154535209732295, 1.27466931826255252411951582982, 2.24546347374920054558505230000, 3.17600717071126202663279003321, 4.44062521739098601708547429922, 5.02876112599571306542433430504, 6.06360202788190937242513294517, 6.48337995131101113686227003838, 7.06932926715160374277744658129, 8.454333341611953025706836856466, 8.868536353120836470189309613967

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