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Properties

Label 2-3872-88.75-c0-0-1
Degree $2$
Conductor $3872$
Sign $0.0219 - 0.999i$
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  + (−1.30 + 0.951i)3-s + (0.500 − 1.53i)9-s + (0.190 + 0.587i)17-s + (0.5 − 0.363i)19-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (1.30 − 0.951i)41-s + 1.61·43-s + (0.309 + 0.951i)49-s + (−0.809 − 0.587i)51-s + (−0.309 + 0.951i)57-s + (0.5 + 0.363i)59-s − 0.618·67-s + (1.30 + 0.951i)73-s + (0.5 − 1.53i)75-s + ⋯
L(s)  = 1  + (−1.30 + 0.951i)3-s + (0.500 − 1.53i)9-s + (0.190 + 0.587i)17-s + (0.5 − 0.363i)19-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (1.30 − 0.951i)41-s + 1.61·43-s + (0.309 + 0.951i)49-s + (−0.809 − 0.587i)51-s + (−0.309 + 0.951i)57-s + (0.5 + 0.363i)59-s − 0.618·67-s + (1.30 + 0.951i)73-s + (0.5 − 1.53i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3872\)    =    \(2^{5} \cdot 11^{2}\)
Sign: $0.0219 - 0.999i$
Analytic conductor: \(1.93237\)
Root analytic conductor: \(1.39010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3872} (2671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3872,\ (\ :0),\ 0.0219 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7383908307\)
\(L(\frac12)\) \(\approx\) \(0.7383908307\)
\(L(1)\) 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 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.61T + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.196409396962537377056976679090, −8.051718782102147034269799119664, −7.28235121253237444136835847260, −6.40456257275428403565794729717, −5.65869692664908462425563826728, −5.29530044953628164443853671495, −4.23443281559646161965441633025, −3.82776802643958777810612639163, −2.53757958295007691175271558561, −1.02671392762518929445375979857, 0.63160312063844911773506738522, 1.70270477977171873244679121242, 2.76391064844183213849282200554, 4.03246293479021643975295774981, 4.92602962460192068984586526441, 5.72133682082126376360895262484, 6.14097983279626670588289387284, 7.00615430887541538212191911658, 7.55480749976502482445946363062, 8.202288729785028763356844294480

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