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Properties

Label 2-3420-3420.1139-c0-0-12
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 + (0.130 + 0.991i)3-s + (−0.707 + 0.707i)4-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.5i)6-s + (−0.923 − 0.382i)8-s + (−0.965 + 0.258i)9-s + (0.130 − 0.991i)10-s + (−0.965 − 1.67i)11-s + (−0.793 − 0.608i)12-s + (0.608 − 1.05i)13-s + (0.382 − 0.923i)15-s − i·16-s + (−0.608 − 0.793i)18-s + i·19-s + (0.965 − 0.258i)20-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (0.130 + 0.991i)3-s + (−0.707 + 0.707i)4-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.5i)6-s + (−0.923 − 0.382i)8-s + (−0.965 + 0.258i)9-s + (0.130 − 0.991i)10-s + (−0.965 − 1.67i)11-s + (−0.793 − 0.608i)12-s + (0.608 − 1.05i)13-s + (0.382 − 0.923i)15-s − i·16-s + (−0.608 − 0.793i)18-s + i·19-s + (0.965 − 0.258i)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

\(L(\frac{1}{2})\) \(\approx\) \(0.7960859287\)
\(L(\frac12)\) \(\approx\) \(0.7960859287\)
\(L(1)\) 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 \( 1 + (-0.130 - 0.991i)T \)
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 \( 1 + (-0.608 + 1.05i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 1.58T + T^{2} \)
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 \( 1 + 1.98iT - T^{2} \)
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

−8.417813573275161747089251985850, −8.193110254003608582452800741098, −7.66625896251294504926527445832, −6.27206296432446868748527467751, −5.62372081336192727348308403322, −5.14979177883727858782639355951, −4.19546440470721653980287038473, −3.45051248215405205490879840770, −3.00742623109314702710638769858, −0.43716062954594850934590561465, 1.29316703067997315386783536775, 2.43706979737001561629219522435, 2.84844931626722532432815480533, 4.17653379268898894662047481877, 4.58822412486273019525767748625, 5.75447035168559390294296229650, 6.61550033000681164908566651480, 7.32090816541338740713026654139, 7.88010918233863539768105324049, 8.882629177164387529449580377658

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