Properties

Label 2-3420-95.94-c0-0-2
Degree $2$
Conductor $3420$
Sign $0.866 - 0.5i$
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  + (0.866 − 0.5i)5-s + 1.73i·7-s + 1.73·11-s − i·17-s − 19-s + 2i·23-s + (0.499 − 0.866i)25-s + (0.866 + 1.49i)35-s − 1.73i·43-s + i·47-s − 1.99·49-s + (1.49 − 0.866i)55-s − 61-s + 1.73i·73-s + 2.99i·77-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)5-s + 1.73i·7-s + 1.73·11-s − i·17-s − 19-s + 2i·23-s + (0.499 − 0.866i)25-s + (0.866 + 1.49i)35-s − 1.73i·43-s + i·47-s − 1.99·49-s + (1.49 − 0.866i)55-s − 61-s + 1.73i·73-s + 2.99i·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(1.70680\)
Root analytic conductor: \(1.30644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (2089, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :0),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.615025585\)
\(L(\frac12)\) \(\approx\) \(1.615025585\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + T \)
good7 \( 1 - 1.73iT - T^{2} \)
11 \( 1 - 1.73T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + iT - T^{2} \)
23 \( 1 - 2iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 2iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.049170273933683397762835063701, −8.424177103573883432802080093178, −7.26270838137862383780736443214, −6.40265830629656190574326049017, −5.81229029049140974111447322922, −5.24886438150428821568872851706, −4.30987823633769286770132987369, −3.20915293388545543579034120039, −2.18697013244615416469418341715, −1.46942692113537739013303843640, 1.10068458968317322435587862502, 2.01473288901324119161482277346, 3.30941705912679492089249100270, 4.13384569196571870009500822415, 4.61055035300875020600042858018, 6.08903252918108283469238454761, 6.54313124171482951218308845041, 6.93461227781443016160920569177, 7.982304580097180821184957457626, 8.750243660524342739062480783682

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