Properties

Label 2-57e2-19.12-c0-0-0
Degree $2$
Conductor $3249$
Sign $0.671 - 0.740i$
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  + (−0.5 + 0.866i)4-s + 7-s + (1.5 + 0.866i)13-s + (−0.499 − 0.866i)16-s + (0.5 − 0.866i)25-s + (−0.5 + 0.866i)28-s − 1.73i·31-s + 1.73i·37-s + (0.5 + 0.866i)43-s + (−1.5 + 0.866i)52-s + (−0.5 + 0.866i)61-s + 0.999·64-s + (−1.5 − 0.866i)67-s + (0.5 + 0.866i)73-s + (1.5 − 0.866i)79-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + 7-s + (1.5 + 0.866i)13-s + (−0.499 − 0.866i)16-s + (0.5 − 0.866i)25-s + (−0.5 + 0.866i)28-s − 1.73i·31-s + 1.73i·37-s + (0.5 + 0.866i)43-s + (−1.5 + 0.866i)52-s + (−0.5 + 0.866i)61-s + 0.999·64-s + (−1.5 − 0.866i)67-s + (0.5 + 0.866i)73-s + (1.5 − 0.866i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3249\)    =    \(3^{2} \cdot 19^{2}\)
Sign: $0.671 - 0.740i$
Analytic conductor: \(1.62146\)
Root analytic conductor: \(1.27336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3249} (1513, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3249,\ (\ :0),\ 0.671 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.340253485\)
\(L(\frac12)\) \(\approx\) \(1.340253485\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−8.840072184102666070232218288755, −8.115951285452561963505383518771, −7.77305045492050339229067379153, −6.66969113749752101259928434276, −6.00817633412880593672152776205, −4.81180736765956577315101229411, −4.33931518699122910180995037902, −3.54018624267618144778190584879, −2.46967940259246759521080510718, −1.27923637187373894372943988321, 1.02962987883543418812191685697, 1.83043379647876163264005803160, 3.27224063039232803522736910679, 4.13738257279103385877408229951, 5.09407706400131017447352677758, 5.53478311431452852529617122078, 6.33087742920474596172874830226, 7.28020870363842873542245277028, 8.164575142244402909451989518430, 8.773531421980081763754388289357

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