Properties

Label 2-3879-3879.1723-c0-0-17
Degree $2$
Conductor $3879$
Sign $-0.988 - 0.149i$
Analytic cond. $1.93587$
Root an. cond. $1.39135$
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  + (−0.623 − 1.07i)2-s + (−0.411 + 0.911i)3-s + (−0.277 + 0.480i)4-s + (0.853 − 1.47i)5-s + (1.24 − 0.124i)6-s − 0.554·8-s + (−0.661 − 0.749i)9-s − 2.12·10-s + (−0.766 − 1.32i)11-s + (−0.323 − 0.450i)12-s + (0.996 + 1.38i)15-s + (0.623 + 1.07i)16-s + (−0.397 + 1.18i)18-s + 0.541·19-s + (0.473 + 0.820i)20-s + ⋯
L(s)  = 1  + (−0.623 − 1.07i)2-s + (−0.411 + 0.911i)3-s + (−0.277 + 0.480i)4-s + (0.853 − 1.47i)5-s + (1.24 − 0.124i)6-s − 0.554·8-s + (−0.661 − 0.749i)9-s − 2.12·10-s + (−0.766 − 1.32i)11-s + (−0.323 − 0.450i)12-s + (0.996 + 1.38i)15-s + (0.623 + 1.07i)16-s + (−0.397 + 1.18i)18-s + 0.541·19-s + (0.473 + 0.820i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3879\)    =    \(3^{2} \cdot 431\)
Sign: $-0.988 - 0.149i$
Analytic conductor: \(1.93587\)
Root analytic conductor: \(1.39135\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3879} (1723, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3879,\ (\ :0),\ -0.988 - 0.149i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6716849845\)
\(L(\frac12)\) \(\approx\) \(0.6716849845\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.411 - 0.911i)T \)
431 \( 1 - T \)
good2 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.853 + 1.47i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 0.541T + T^{2} \)
23 \( 1 + (-0.998 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.124 - 0.215i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + 1.93T + T^{2} \)
59 \( 1 + (-0.661 + 1.14i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.921 + 1.59i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.494179675373031152300529276336, −8.264162258775954745375009821178, −6.43021408824953372599637207999, −5.90675374171730632090594426410, −5.08501369805838576710086786696, −4.63792792757333550883386218278, −3.32682242870274011053825894990, −2.70970391156150697947851995434, −1.38779421545010811977358713394, −0.49912985818998422456694566447, 1.68736240672744617999145372765, 2.56803639100031207697952374888, 3.33840060001997138542222899835, 5.17610264025641457813820015493, 5.54705212268373183587330414200, 6.44899602509734808234985564645, 6.94733060483503010884388363265, 7.41698984382770605940203698317, 7.82643127599484744341856424377, 8.928634711983882584964807677067

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