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Properties

Label 2-3872-8.3-c0-0-2
Degree $2$
Conductor $3872$
Sign $1$
Analytic cond. $1.93237$
Root an. cond. $1.39010$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.61·3-s + 1.61·9-s − 0.618·17-s + 0.618·19-s + 25-s + 27-s + 1.61·41-s − 1.61·43-s + 49-s − 1.00·51-s + 1.00·57-s − 0.618·59-s − 0.618·67-s + 1.61·73-s + 1.61·75-s − 1.61·83-s − 1.61·89-s − 1.61·97-s − 1.61·107-s + 0.618·113-s + ⋯
L(s)  = 1  + 1.61·3-s + 1.61·9-s − 0.618·17-s + 0.618·19-s + 25-s + 27-s + 1.61·41-s − 1.61·43-s + 49-s − 1.00·51-s + 1.00·57-s − 0.618·59-s − 0.618·67-s + 1.61·73-s + 1.61·75-s − 1.61·83-s − 1.61·89-s − 1.61·97-s − 1.61·107-s + 0.618·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3872\)    =    \(2^{5} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1.93237\)
Root analytic conductor: \(1.39010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3872} (3631, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3872,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.326243617\)
\(L(\frac12)\) \(\approx\) \(2.326243617\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 - 1.61T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 0.618T + T^{2} \)
19 \( 1 - 0.618T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.61T + T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 0.618T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.61T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.61T + T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 + 1.61T + T^{2} \)
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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

−8.595933792893190839291034256669, −8.110575638424033873597614878888, −7.29865953508459547599061223566, −6.75908906855448108921041416278, −5.65664980831310091699390725910, −4.65148841785606274880056595812, −3.91332319742852020579738943377, −3.04991358719234306869605469447, −2.44782373792378154423429403954, −1.38245598383719985859981152526, 1.38245598383719985859981152526, 2.44782373792378154423429403954, 3.04991358719234306869605469447, 3.91332319742852020579738943377, 4.65148841785606274880056595812, 5.65664980831310091699390725910, 6.75908906855448108921041416278, 7.29865953508459547599061223566, 8.110575638424033873597614878888, 8.595933792893190839291034256669

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