Properties

Label 2-63e2-7.5-c0-0-3
Degree $2$
Conductor $3969$
Sign $-0.895 - 0.444i$
Analytic cond. $1.98078$
Root an. cond. $1.40740$
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  + (0.866 − 1.5i)2-s + (−1 − 1.73i)4-s − 1.73·8-s + (−0.866 − 1.5i)11-s + (−0.5 + 0.866i)16-s − 3·22-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)37-s + 43-s + (−1.73 + 3i)44-s − 1.73·50-s + (−0.866 − 1.5i)53-s − 0.999·64-s + (0.5 + 0.866i)67-s − 1.73·71-s + ⋯
L(s)  = 1  + (0.866 − 1.5i)2-s + (−1 − 1.73i)4-s − 1.73·8-s + (−0.866 − 1.5i)11-s + (−0.5 + 0.866i)16-s − 3·22-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)37-s + 43-s + (−1.73 + 3i)44-s − 1.73·50-s + (−0.866 − 1.5i)53-s − 0.999·64-s + (0.5 + 0.866i)67-s − 1.73·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.895 - 0.444i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (1783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.541248885\)
\(L(\frac12)\) \(\approx\) \(1.541248885\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−8.398971281812037569437184215340, −7.61999557829390956379270184286, −6.36843082560268455770981563537, −5.71474397797089338983141582548, −5.03739806561117057335019659410, −4.22865586431004958468657100943, −3.35897459107896017395468529004, −2.81875103999286883267258364009, −1.87665003720305507883228671364, −0.62284213264159484691769096461, 1.93930073926760297653388806448, 3.09847668027719573965562988434, 4.11597796690524910150070249510, 4.69596103101586496822016335487, 5.41058071613123887342595419718, 6.01645291934499231668423535754, 6.93547353466642123682207585027, 7.48451470406503086417831299777, 7.84651258200414070771364160517, 8.836991074674382989909616313813

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