Properties

Label 2-57e2-19.14-c0-0-1
Degree $2$
Conductor $3249$
Sign $0.304 + 0.952i$
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  + (0.766 − 0.642i)4-s + (−0.5 − 0.866i)7-s + (1.70 − 0.300i)13-s + (0.173 − 0.984i)16-s + (−0.173 − 0.984i)25-s + (−0.939 − 0.342i)28-s + (−1.5 + 0.866i)31-s + 1.73i·37-s + (−0.766 − 0.642i)43-s + (1.11 − 1.32i)52-s + (0.766 − 0.642i)61-s + (−0.500 − 0.866i)64-s + (0.592 − 1.62i)67-s + (−0.173 + 0.984i)73-s + (1.70 + 0.300i)79-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)4-s + (−0.5 − 0.866i)7-s + (1.70 − 0.300i)13-s + (0.173 − 0.984i)16-s + (−0.173 − 0.984i)25-s + (−0.939 − 0.342i)28-s + (−1.5 + 0.866i)31-s + 1.73i·37-s + (−0.766 − 0.642i)43-s + (1.11 − 1.32i)52-s + (0.766 − 0.642i)61-s + (−0.500 − 0.866i)64-s + (0.592 − 1.62i)67-s + (−0.173 + 0.984i)73-s + (1.70 + 0.300i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3249\)    =    \(3^{2} \cdot 19^{2}\)
Sign: $0.304 + 0.952i$
Analytic conductor: \(1.62146\)
Root analytic conductor: \(1.27336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3249} (3187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3249,\ (\ :0),\ 0.304 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.489551519\)
\(L(\frac12)\) \(\approx\) \(1.489551519\)
\(L(1)\) 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 \( 1 + (-0.766 + 0.642i)T^{2} \)
5 \( 1 + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.70 + 0.300i)T + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.766 + 0.642i)T^{2} \)
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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

−8.596160179251757025892925941648, −7.947444107414515212066435895506, −6.90501796400143512990243668982, −6.56906821600786214942760333741, −5.79958839474875599971165106542, −4.96002134659983117743639536372, −3.78601309154828107251850172803, −3.21427440428828665920101455504, −1.91832411437256368112242723041, −0.932078019798170067052399531276, 1.61075690811809028187190164740, 2.50788933762156103365399524622, 3.51161516678686829706705452382, 3.96993315372447799622206024359, 5.49102795903079912525543414623, 5.98007554947056407559081341054, 6.72187047388390880609152760521, 7.47830657588383029561815223714, 8.254962194027397371482410799334, 8.979122819354706931894465151542

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