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Properties

Label 2-3871-553.473-c0-0-12
Degree $2$
Conductor $3871$
Sign $0.991 - 0.126i$
Analytic cond. $1.93188$
Root an. cond. $1.38992$
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  + (0.809 + 1.40i)2-s + (−0.809 + 1.40i)4-s + (−0.309 − 0.535i)5-s − 0.999·8-s + (−0.5 − 0.866i)9-s + (0.499 − 0.866i)10-s + (0.809 − 1.40i)11-s − 1.61·13-s + (0.809 − 1.40i)18-s + (−0.309 − 0.535i)19-s + 0.999·20-s + 2.61·22-s + (−0.309 − 0.535i)23-s + (0.309 − 0.535i)25-s + (−1.30 − 2.26i)26-s + ⋯
L(s)  = 1  + (0.809 + 1.40i)2-s + (−0.809 + 1.40i)4-s + (−0.309 − 0.535i)5-s − 0.999·8-s + (−0.5 − 0.866i)9-s + (0.499 − 0.866i)10-s + (0.809 − 1.40i)11-s − 1.61·13-s + (0.809 − 1.40i)18-s + (−0.309 − 0.535i)19-s + 0.999·20-s + 2.61·22-s + (−0.309 − 0.535i)23-s + (0.309 − 0.535i)25-s + (−1.30 − 2.26i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3871\)    =    \(7^{2} \cdot 79\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(1.93188\)
Root analytic conductor: \(1.38992\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3871} (3791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3871,\ (\ :0),\ 0.991 - 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.471331269\)
\(L(\frac12)\) \(\approx\) \(1.471331269\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
79 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.61T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 0.618T + T^{2} \)
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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

−8.486359616237138271319600376359, −7.86230624515440879192604485042, −7.02147080841930169026492813216, −6.31073830660997941292833676860, −5.90171824620450564999641349229, −4.94998973327039984275824247256, −4.35563152644912492675661090504, −3.59274002488400520036042299138, −2.59515710960398674045930546840, −0.62531970869581775371546194955, 1.62386949582503797664970533366, 2.30479749010096875475985333883, 3.07515285098843577488421230566, 3.93870697455737173873386735713, 4.84491414526951794160399749363, 5.07501538190120198681140174816, 6.33119099214270227200999369702, 7.29992834387653379749757531246, 7.69185436486764516030636116355, 8.918515860996151245225233778814

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