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Properties

Label 2-3420-3420.2279-c0-0-11
Degree $2$
Conductor $3420$
Sign $0.766 + 0.642i$
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  + (0.130 + 0.991i)2-s + (−0.608 − 0.793i)3-s + (−0.965 + 0.258i)4-s + (0.866 − 0.5i)5-s + (0.707 − 0.707i)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 + (0.793 + 0.608i)12-s + (−0.130 − 0.226i)13-s + (−0.923 − 0.382i)15-s + (0.866 − 0.5i)16-s + (−0.991 − 0.130i)18-s − i·19-s + (−0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.608 − 0.793i)3-s + (−0.965 + 0.258i)4-s + (0.866 − 0.5i)5-s + (0.707 − 0.707i)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 + (0.793 + 0.608i)12-s + (−0.130 − 0.226i)13-s + (−0.923 − 0.382i)15-s + (0.866 − 0.5i)16-s + (−0.991 − 0.130i)18-s − i·19-s + (−0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.766 + 0.642i$
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: \((2,\ 3420,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.012080222\)
\(L(\frac12)\) \(\approx\) \(1.012080222\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.130 - 0.991i)T \)
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

−8.551866495822772953077943189057, −7.908709359152454533834940462901, −6.87476517716202356123578688064, −6.62034757165040573255097929264, −5.64453947798927230582107722113, −5.26618727610184782261134691462, −4.48472751705546701325641483525, −3.18228999702560449612147326128, −1.91909140064854229525746504538, −0.66357102015158349358907311905, 1.39289513815373358368197000347, 2.39836636892107978870594224145, 3.43534726815659075525077427257, 4.08726426377000348961679484068, 5.05409809421712337525693007847, 5.60908436430732942804065685628, 6.36791016221368135529796660700, 7.25176632249940555389090032628, 8.551287669426615980787598022588, 9.156764868638427495150829208465

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