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Properties

Label 2-3871-553.135-c0-0-0
Degree $2$
Conductor $3871$
Sign $0.197 + 0.980i$
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  − 2-s + 8-s + (−0.5 − 0.866i)9-s + 11-s − 16-s + (0.5 + 0.866i)18-s − 22-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (0.5 + 0.866i)46-s + (−0.5 + 0.866i)50-s + 64-s + (−0.5 − 0.866i)67-s + (−0.5 − 0.866i)72-s + (−0.5 + 0.866i)79-s + ⋯
L(s)  = 1  − 2-s + 8-s + (−0.5 − 0.866i)9-s + 11-s − 16-s + (0.5 + 0.866i)18-s − 22-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (0.5 + 0.866i)46-s + (−0.5 + 0.866i)50-s + 64-s + (−0.5 − 0.866i)67-s + (−0.5 − 0.866i)72-s + (−0.5 + 0.866i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3871\)    =    \(7^{2} \cdot 79\)
Sign: $0.197 + 0.980i$
Analytic conductor: \(1.93188\)
Root analytic conductor: \(1.38992\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3871} (1794, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3871,\ (\ :0),\ 0.197 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5973903059\)
\(L(\frac12)\) \(\approx\) \(0.5973903059\)
\(L(1)\) 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 \( 1 + T + T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.522372487556273877104176953120, −8.115482191476161929346081374928, −7.02665043052165480175275374324, −6.52621718353394441329177814913, −5.68150886757885049577429223343, −4.54550075958029585764480444184, −3.96565406353886789500033884056, −2.87766952058025046852395279939, −1.65713397760967930280565207678, −0.55533557954996044871708081712, 1.21331496583476193237171108161, 2.10030371001287599669194364282, 3.35403926856001436298078577049, 4.27745540528511277394699875975, 5.08825842473020643479186580872, 5.87933705701696419142990543403, 6.89168190706434506016082826635, 7.55696145669105068210685739774, 8.167473156230140908428166429159, 8.920929522333761339245498462185

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