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Properties

Label 2-3871-79.78-c0-0-3
Degree $2$
Conductor $3871$
Sign $1$
Analytic cond. $1.93188$
Root an. cond. $1.38992$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s + 1.61·4-s − 0.618·5-s − 8-s + 9-s + 1.00·10-s − 1.61·11-s + 1.61·13-s − 1.61·18-s − 0.618·19-s − 1.00·20-s + 2.61·22-s + 0.618·23-s − 0.618·25-s − 2.61·26-s + 1.61·31-s + 32-s + 1.61·36-s + 1.00·38-s + 0.618·40-s − 2.61·44-s − 0.618·45-s − 1.00·46-s + 0.999·50-s + 2.61·52-s + 1.00·55-s − 2.61·62-s + ⋯
L(s)  = 1  − 1.61·2-s + 1.61·4-s − 0.618·5-s − 8-s + 9-s + 1.00·10-s − 1.61·11-s + 1.61·13-s − 1.61·18-s − 0.618·19-s − 1.00·20-s + 2.61·22-s + 0.618·23-s − 0.618·25-s − 2.61·26-s + 1.61·31-s + 32-s + 1.61·36-s + 1.00·38-s + 0.618·40-s − 2.61·44-s − 0.618·45-s − 1.00·46-s + 0.999·50-s + 2.61·52-s + 1.00·55-s − 2.61·62-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3871\)    =    \(7^{2} \cdot 79\)
Sign: $1$
Analytic conductor: \(1.93188\)
Root analytic conductor: \(1.38992\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3871} (2843, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3871,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5179639002\)
\(L(\frac12)\) \(\approx\) \(0.5179639002\)
\(L(1)\) 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 \( 1 - T \)
good2 \( 1 + 1.61T + T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 + 0.618T + T^{2} \)
11 \( 1 + 1.61T + T^{2} \)
13 \( 1 - 1.61T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 0.618T + T^{2} \)
23 \( 1 - 0.618T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.61T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 0.618T + T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 - 1.61T + T^{2} \)
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

−8.483752216984281511831028398115, −8.087689005730467510298560007433, −7.52444368932463856971907750757, −6.76349931858681190969506761185, −6.02293828762766177802572178255, −4.84582754058516930157914867655, −4.00197664824122313983798470526, −2.90615668789877121469611312661, −1.86071031013832090378576423191, −0.795087921720937410396009565130, 0.795087921720937410396009565130, 1.86071031013832090378576423191, 2.90615668789877121469611312661, 4.00197664824122313983798470526, 4.84582754058516930157914867655, 6.02293828762766177802572178255, 6.76349931858681190969506761185, 7.52444368932463856971907750757, 8.087689005730467510298560007433, 8.483752216984281511831028398115

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