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Properties

Label 2-3420-3420.1139-c0-0-15
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  + (0.793 − 0.608i)2-s + (0.991 − 0.130i)3-s + (0.258 − 0.965i)4-s + (−0.866 − 0.5i)5-s + (0.707 − 0.707i)6-s + (−0.382 − 0.923i)8-s + (0.965 − 0.258i)9-s + (−0.991 + 0.130i)10-s + (−0.965 − 1.67i)11-s + (0.130 − 0.991i)12-s + (−0.793 + 1.37i)13-s + (−0.923 − 0.382i)15-s + (−0.866 − 0.499i)16-s + (0.608 − 0.793i)18-s − i·19-s + (−0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)2-s + (0.991 − 0.130i)3-s + (0.258 − 0.965i)4-s + (−0.866 − 0.5i)5-s + (0.707 − 0.707i)6-s + (−0.382 − 0.923i)8-s + (0.965 − 0.258i)9-s + (−0.991 + 0.130i)10-s + (−0.965 − 1.67i)11-s + (0.130 − 0.991i)12-s + (−0.793 + 1.37i)13-s + (−0.923 − 0.382i)15-s + (−0.866 − 0.499i)16-s + (0.608 − 0.793i)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: $\chi_{3420} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :0),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.083537776\)
\(L(\frac12)\) \(\approx\) \(2.083537776\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.793 + 0.608i)T \)
3 \( 1 + (-0.991 + 0.130i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.793 - 1.37i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 1.21T + T^{2} \)
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 \( 1 + 0.261iT - T^{2} \)
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

−8.714741839737108623659313321480, −7.72829670127337784611151086629, −7.09942585001661623127998659357, −6.19814505398968739883181438166, −5.11886167966613857304073863656, −4.48738117322833908521155344309, −3.71876518707225357180094805345, −2.94498887402321492475303260403, −2.20496094170765588571483042032, −0.811112552393919039379367480379, 2.21405879764534236248694964097, 2.84190455339914079449166631482, 3.63284430255391526509416888372, 4.50311667805950531292391341658, 5.00421181687238943659346477990, 6.10381937348911185954383914810, 7.20673252578421649666402418831, 7.63432265714274344678481964885, 7.86493445920096574656388695641, 8.744958647138070221444216129264

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