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Properties

Label 2-3872-11.2-c0-0-2
Degree $2$
Conductor $3872$
Sign $0.396 - 0.917i$
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  + (−0.309 + 0.951i)9-s + (1.34 + 0.437i)13-s + (−1.34 + 0.437i)17-s + (0.809 − 0.587i)25-s + (−0.831 + 1.14i)29-s + (1.61 + 1.17i)37-s + (−0.831 − 1.14i)41-s + (0.309 + 0.951i)49-s + (−0.618 + 1.90i)53-s + (−1.34 + 0.437i)61-s + (0.831 − 1.14i)73-s + (−0.809 − 0.587i)81-s + 2·89-s + (1.34 + 0.437i)101-s + 1.41i·109-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)9-s + (1.34 + 0.437i)13-s + (−1.34 + 0.437i)17-s + (0.809 − 0.587i)25-s + (−0.831 + 1.14i)29-s + (1.61 + 1.17i)37-s + (−0.831 − 1.14i)41-s + (0.309 + 0.951i)49-s + (−0.618 + 1.90i)53-s + (−1.34 + 0.437i)61-s + (0.831 − 1.14i)73-s + (−0.809 − 0.587i)81-s + 2·89-s + (1.34 + 0.437i)101-s + 1.41i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3872\)    =    \(2^{5} \cdot 11^{2}\)
Sign: $0.396 - 0.917i$
Analytic conductor: \(1.93237\)
Root analytic conductor: \(1.39010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3872} (3137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3872,\ (\ :0),\ 0.396 - 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.181143037\)
\(L(\frac12)\) \(\approx\) \(1.181143037\)
\(L(1)\) 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 \( 1 + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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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

−8.887780150439104764072802091108, −8.113618414792452379323727301467, −7.39586780128980637741165760825, −6.43308909227761896426253641132, −6.00906011236270065415402632883, −4.90954706395442221425906888917, −4.34400123263516743458687488414, −3.34233182067619751519590170843, −2.36303670167254789371035265555, −1.40533591163448978259427146473, 0.70173201103067781215567915997, 2.03397964865895315910821092875, 3.14439973003531019032530336506, 3.81868602102900334185343000195, 4.69049757274454529232047121730, 5.69732314887534612358386864154, 6.32222397794052967917380300566, 6.90190445064745387079403033776, 7.906863579373144942463864031026, 8.568430763925816910904685229750

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