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Properties

Label 2-57e2-19.2-c0-0-1
Degree $2$
Conductor $3249$
Sign $0.755 + 0.654i$
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  + (−0.939 − 0.342i)4-s + (−0.5 − 0.866i)7-s + (1.11 + 1.32i)13-s + (0.766 + 0.642i)16-s + (−0.766 + 0.642i)25-s + (0.173 + 0.984i)28-s + (1.5 − 0.866i)31-s − 1.73i·37-s + (0.939 − 0.342i)43-s + (−0.592 − 1.62i)52-s + (−0.939 − 0.342i)61-s + (−0.500 − 0.866i)64-s + (1.70 − 0.300i)67-s + (−0.766 − 0.642i)73-s + (1.11 − 1.32i)79-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)4-s + (−0.5 − 0.866i)7-s + (1.11 + 1.32i)13-s + (0.766 + 0.642i)16-s + (−0.766 + 0.642i)25-s + (0.173 + 0.984i)28-s + (1.5 − 0.866i)31-s − 1.73i·37-s + (0.939 − 0.342i)43-s + (−0.592 − 1.62i)52-s + (−0.939 − 0.342i)61-s + (−0.500 − 0.866i)64-s + (1.70 − 0.300i)67-s + (−0.766 − 0.642i)73-s + (1.11 − 1.32i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3249\)    =    \(3^{2} \cdot 19^{2}\)
Sign: $0.755 + 0.654i$
Analytic conductor: \(1.62146\)
Root analytic conductor: \(1.27336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3249} (838, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3249,\ (\ :0),\ 0.755 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9483591549\)
\(L(\frac12)\) \(\approx\) \(0.9483591549\)
\(L(1)\) 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 \( 1 + (0.939 + 0.342i)T^{2} \)
5 \( 1 + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.11 - 1.32i)T + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (-1.70 + 0.300i)T + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T^{2} \)
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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

−8.968289721068711299712485143256, −8.042555111643154883201939448610, −7.29194834545212748214111903666, −6.35533463126001153502404429857, −5.85577589157470201961610611745, −4.73395226930666640344412367605, −4.01738822260711280600167079204, −3.56201689384055846235084576853, −1.98731494336780366780078782333, −0.78635379002860614865184952505, 1.02088597221154686428042220313, 2.70621255664163314640188955898, 3.30919404038484838734620340401, 4.24670027893042456925295100993, 5.13535687501853324705995001852, 5.87620340573936735745718218403, 6.47035046285359623593731416176, 7.70350660474099513531056265113, 8.397453474134345971809922230166, 8.662517434618304374865069933117

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