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Properties

Label 2-3871-553.394-c0-0-2
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  + (−0.309 + 0.535i)2-s + (0.309 + 0.535i)4-s + (0.587 − 1.01i)5-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (0.363 + 0.629i)10-s + (0.309 + 0.535i)11-s − 1.90·13-s + (−0.309 − 0.535i)18-s + (−0.587 + 1.01i)19-s + 0.726·20-s − 0.381·22-s + (−0.809 + 1.40i)23-s + (−0.190 − 0.330i)25-s + (0.587 − 1.01i)26-s + ⋯
L(s)  = 1  + (−0.309 + 0.535i)2-s + (0.309 + 0.535i)4-s + (0.587 − 1.01i)5-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (0.363 + 0.629i)10-s + (0.309 + 0.535i)11-s − 1.90·13-s + (−0.309 − 0.535i)18-s + (−0.587 + 1.01i)19-s + 0.726·20-s − 0.381·22-s + (−0.809 + 1.40i)23-s + (−0.190 − 0.330i)25-s + (0.587 − 1.01i)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: $\chi_{3871} (1500, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3871,\ (\ :0),\ -0.991 - 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5883718699\)
\(L(\frac12)\) \(\approx\) \(0.5883718699\)
\(L(1)\) 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 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.587 + 1.01i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.90T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.587 - 1.01i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.951 + 1.64i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.587 - 1.01i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.951 - 1.64i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.17T + T^{2} \)
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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

−9.010286650240881469245808307408, −8.000538019702537795413740879107, −7.75666969703567421998424654989, −7.01569657270299301656052417006, −5.93055160150716100196878510974, −5.45404624587822777948548553846, −4.64047449668942901948610226139, −3.70180779874214079730385429738, −2.38355283344323639850196272887, −1.88508075035991915843235837557, 0.31705660321478611486902155074, 1.91412241554188185707797258094, 2.69710203150533570807574634396, 3.15188964138775866271236113610, 4.51400341303257418321791668789, 5.46998080295150206218241755554, 6.26091608002606603774798273027, 6.71197606558329343146262586315, 7.33175796000469743114309491619, 8.714470601243538749982708294666

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