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Properties

Label 2-3879-3879.1723-c0-0-15
Degree $2$
Conductor $3879$
Sign $0.826 - 0.563i$
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  + (0.900 + 1.56i)2-s + (−0.998 − 0.0498i)3-s + (−1.12 + 1.94i)4-s + (0.969 − 1.67i)5-s + (−0.822 − 1.60i)6-s − 2.24·8-s + (0.995 + 0.0995i)9-s + 3.49·10-s + (−0.173 − 0.300i)11-s + (1.21 − 1.88i)12-s + (−1.05 + 1.62i)15-s + (−0.900 − 1.56i)16-s + (0.741 + 1.64i)18-s − 0.248·19-s + (2.17 + 3.77i)20-s + ⋯
L(s)  = 1  + (0.900 + 1.56i)2-s + (−0.998 − 0.0498i)3-s + (−1.12 + 1.94i)4-s + (0.969 − 1.67i)5-s + (−0.822 − 1.60i)6-s − 2.24·8-s + (0.995 + 0.0995i)9-s + 3.49·10-s + (−0.173 − 0.300i)11-s + (1.21 − 1.88i)12-s + (−1.05 + 1.62i)15-s + (−0.900 − 1.56i)16-s + (0.741 + 1.64i)18-s − 0.248·19-s + (2.17 + 3.77i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3879\)    =    \(3^{2} \cdot 431\)
Sign: $0.826 - 0.563i$
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: \((2,\ 3879,\ (\ :0),\ 0.826 - 0.563i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.557113948\)
\(L(\frac12)\) \(\approx\) \(1.557113948\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.998 + 0.0498i)T \)
431 \( 1 - T \)
good2 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.969 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 0.248T + T^{2} \)
23 \( 1 + (-0.853 + 1.47i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.980 + 1.69i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 1.84T + T^{2} \)
59 \( 1 + (0.995 - 1.72i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.318 - 0.551i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.660814268294384256158109744202, −7.77575358070542355219023013091, −7.04993988567895845867964705206, −6.08277667865039686645764083041, −5.81451957564521156666569562779, −5.24178724136583103364829392134, −4.44122119571179274579817502469, −4.13288583889870776447375616188, −2.30161238450186544220121562077, −0.77790120024199249251225644423, 1.46832475405711522804558167785, 2.14234563652839567967027664607, 3.18632125713221898942291757801, 3.65803258203311886841521906022, 4.87915272730429935578261512251, 5.42470642020903207696261583004, 6.08786111555682801330897734915, 6.83870934909314694446447017400, 7.49046520885693405885929235705, 9.240540116072960105660086756957

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