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Properties

Label 2-3420-3420.2279-c0-0-14
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  + (0.382 + 0.923i)2-s + (0.793 − 0.608i)3-s + (−0.707 + 0.707i)4-s + (0.866 − 0.5i)5-s + (0.866 + 0.499i)6-s + (−0.923 − 0.382i)8-s + (0.258 − 0.965i)9-s + (0.793 + 0.608i)10-s + (0.258 − 0.448i)11-s + (−0.130 + 0.991i)12-s + (−0.991 − 1.71i)13-s + (0.382 − 0.923i)15-s − i·16-s + (0.991 − 0.130i)18-s + i·19-s + (−0.258 + 0.965i)20-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (0.793 − 0.608i)3-s + (−0.707 + 0.707i)4-s + (0.866 − 0.5i)5-s + (0.866 + 0.499i)6-s + (−0.923 − 0.382i)8-s + (0.258 − 0.965i)9-s + (0.793 + 0.608i)10-s + (0.258 − 0.448i)11-s + (−0.130 + 0.991i)12-s + (−0.991 − 1.71i)13-s + (0.382 − 0.923i)15-s − i·16-s + (0.991 − 0.130i)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: $\chi_{3420} (2279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :0),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.052381629\)
\(L(\frac12)\) \(\approx\) \(2.052381629\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 - 0.923i)T \)
3 \( 1 + (-0.793 + 0.608i)T \)
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 \( 1 + (0.991 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - 0.261T + T^{2} \)
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 \( 1 - 1.21iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.529113641590590738917242928475, −7.935375710492508123012860491503, −7.43147239985886246105709303804, −6.43921692790206617012890676495, −5.82489816707380292698674038020, −5.21785632960138540158185034732, −4.17699258230479938560586990194, −3.18896976778315940965311952195, −2.45956764307783216305008172799, −1.00580975551526524686040598102, 1.81676512848845884097563228608, 2.28213854833632579213309037344, 3.13132688118669066154764641367, 4.13093712610696227973751418849, 4.73707992534701082624087801073, 5.44743063320973414334714180814, 6.65036541050305367525075605203, 7.13790835690176364183852358527, 8.441061801561844690063968155839, 9.154395794747765080080388657728

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