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Properties

Label 2-3420-3420.1139-c0-0-6
Degree $2$
Conductor $3420$
Sign $0.984 - 0.173i$
Analytic cond. $1.70680$
Root an. cond. $1.30644$
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  + (−0.923 − 0.382i)2-s + (−0.608 + 0.793i)3-s + (0.707 + 0.707i)4-s + (0.866 + 0.5i)5-s + (0.866 − 0.499i)6-s + (−0.382 − 0.923i)8-s + (−0.258 − 0.965i)9-s + (−0.608 − 0.793i)10-s + (−0.258 − 0.448i)11-s + (−0.991 + 0.130i)12-s + (0.130 − 0.226i)13-s + (−0.923 + 0.382i)15-s + i·16-s + (−0.130 + 0.991i)18-s − i·19-s + (0.258 + 0.965i)20-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (−0.608 + 0.793i)3-s + (0.707 + 0.707i)4-s + (0.866 + 0.5i)5-s + (0.866 − 0.499i)6-s + (−0.382 − 0.923i)8-s + (−0.258 − 0.965i)9-s + (−0.608 − 0.793i)10-s + (−0.258 − 0.448i)11-s + (−0.991 + 0.130i)12-s + (0.130 − 0.226i)13-s + (−0.923 + 0.382i)15-s + i·16-s + (−0.130 + 0.991i)18-s − i·19-s + (0.258 + 0.965i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(1.70680\)
Root analytic conductor: \(1.30644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7785367698\)
\(L(\frac12)\) \(\approx\) \(0.7785367698\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 0.382i)T \)
3 \( 1 + (0.608 - 0.793i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.130 + 0.226i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.98T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + 1.58iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.992699401468730298040973138879, −8.338217650728594079549776415278, −7.27276224461893349538513627498, −6.56530986056198345683969161222, −5.91473453522791689940282937839, −5.11704902855223877469380295909, −3.99214030190157319808742717337, −3.07266751960508398699671453943, −2.33487889964583394797937349525, −0.867919277408200991660004542321, 1.01294186525285914989552474147, 1.86622510072421525712307288242, 2.65720815492108101038191854309, 4.46041569093696519634866720360, 5.33956764356529257449070530065, 5.99802567339021514993699720403, 6.44954504465441876421322438538, 7.38777926485293254964434711421, 7.946267767920676883708449916615, 8.659056487088697167205886436006

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