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Properties

Label 2-3872-968.115-c0-0-0
Degree $2$
Conductor $3872$
Sign $0.704 - 0.709i$
Analytic cond. $1.93237$
Root an. cond. $1.39010$
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  + (1.25 + 0.910i)3-s + (0.431 + 1.32i)9-s + (−0.0855 − 0.996i)11-s + (1.74 + 0.859i)17-s + (0.0798 − 0.151i)19-s + (0.696 − 0.717i)25-s + (−0.190 + 0.584i)27-s + (0.799 − 1.32i)33-s + (−1.38 − 0.495i)41-s + (−1.29 + 1.48i)43-s + (0.516 + 0.856i)49-s + (1.40 + 2.66i)51-s + (0.237 − 0.116i)57-s + (0.683 − 0.243i)59-s + (−1.55 + 0.455i)67-s + ⋯
L(s)  = 1  + (1.25 + 0.910i)3-s + (0.431 + 1.32i)9-s + (−0.0855 − 0.996i)11-s + (1.74 + 0.859i)17-s + (0.0798 − 0.151i)19-s + (0.696 − 0.717i)25-s + (−0.190 + 0.584i)27-s + (0.799 − 1.32i)33-s + (−1.38 − 0.495i)41-s + (−1.29 + 1.48i)43-s + (0.516 + 0.856i)49-s + (1.40 + 2.66i)51-s + (0.237 − 0.116i)57-s + (0.683 − 0.243i)59-s + (−1.55 + 0.455i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3872\)    =    \(2^{5} \cdot 11^{2}\)
Sign: $0.704 - 0.709i$
Analytic conductor: \(1.93237\)
Root analytic conductor: \(1.39010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3872} (3503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3872,\ (\ :0),\ 0.704 - 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.098828171\)
\(L(\frac12)\) \(\approx\) \(2.098828171\)
\(L(1)\) 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 \( 1 + (0.0855 + 0.996i)T \)
good3 \( 1 + (-1.25 - 0.910i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.696 + 0.717i)T^{2} \)
7 \( 1 + (-0.516 - 0.856i)T^{2} \)
13 \( 1 + (-0.198 - 0.980i)T^{2} \)
17 \( 1 + (-1.74 - 0.859i)T + (0.610 + 0.791i)T^{2} \)
19 \( 1 + (-0.0798 + 0.151i)T + (-0.564 - 0.825i)T^{2} \)
23 \( 1 + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (0.0285 - 0.999i)T^{2} \)
31 \( 1 + (0.870 + 0.491i)T^{2} \)
37 \( 1 + (-0.993 + 0.113i)T^{2} \)
41 \( 1 + (1.38 + 0.495i)T + (0.774 + 0.633i)T^{2} \)
43 \( 1 + (1.29 - 1.48i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (0.254 - 0.967i)T^{2} \)
53 \( 1 + (-0.0855 + 0.996i)T^{2} \)
59 \( 1 + (-0.683 + 0.243i)T + (0.774 - 0.633i)T^{2} \)
61 \( 1 + (0.362 + 0.931i)T^{2} \)
67 \( 1 + (1.55 - 0.455i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.941 - 0.336i)T^{2} \)
73 \( 1 + (0.100 + 0.498i)T + (-0.921 + 0.389i)T^{2} \)
79 \( 1 + (0.466 + 0.884i)T^{2} \)
83 \( 1 + (0.614 - 0.0704i)T + (0.974 - 0.226i)T^{2} \)
89 \( 1 + (0.282 - 1.96i)T + (-0.959 - 0.281i)T^{2} \)
97 \( 1 + (-0.0526 - 0.0222i)T + (0.696 + 0.717i)T^{2} \)
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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

−8.632307090768507216523028012127, −8.250193143162000944546242283681, −7.62965792862559665537273852143, −6.51797143304972866582115791978, −5.67301802582601766918792535891, −4.86001941190505896675027565306, −3.93713965444812598830810390589, −3.28387578298978949497134580313, −2.74382725622103481946497863980, −1.38641621338907185013499016185, 1.26504912898755318526177097343, 2.07163742079041253326576193788, 3.05464037221820705874749948846, 3.57462880689048540547153251559, 4.83955171355713846229992763088, 5.54297516892521503831729061705, 6.75966884236119482349628280699, 7.25972541848680278769392872455, 7.72135410211519533402015845136, 8.530928427437506503868224493222

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