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Properties

Label 2-3420-19.11-c1-0-8
Degree $2$
Conductor $3420$
Sign $0.910 + 0.412i$
Analytic cond. $27.3088$
Root an. cond. $5.22578$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)5-s − 5·7-s − 6·11-s + (0.5 + 0.866i)13-s + (−1 + 1.73i)17-s + (−4 + 1.73i)19-s + (−0.499 − 0.866i)25-s + (−4 − 6.92i)29-s + 9·31-s + (2.5 − 4.33i)35-s + 5·37-s + (−5 + 8.66i)41-s + (−2.5 + 4.33i)43-s + (−6 − 10.3i)47-s + 18·49-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s − 1.88·7-s − 1.80·11-s + (0.138 + 0.240i)13-s + (−0.242 + 0.420i)17-s + (−0.917 + 0.397i)19-s + (−0.0999 − 0.173i)25-s + (−0.742 − 1.28i)29-s + 1.61·31-s + (0.422 − 0.731i)35-s + 0.821·37-s + (−0.780 + 1.35i)41-s + (−0.381 + 0.660i)43-s + (−0.875 − 1.51i)47-s + 2.57·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (3241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5603406570\)
\(L(\frac12)\) \(\approx\) \(0.5603406570\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (4 - 1.73i)T \)
good7 \( 1 + 5T + 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4 + 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9T + 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.354891409095107949556802386116, −7.935463802319255451181594252111, −6.90751800700937748162203686844, −6.33731819351201263539181444856, −5.77202353224747177728029086350, −4.63835219265854490857768282590, −3.73423901720739683063539607454, −2.92515960737218164420344351291, −2.28391039520483982038190783575, −0.31401068046846057921859911035, 0.53698053713868778623731720929, 2.39015665407417866176094500973, 3.01793696084613277761195233955, 3.87505834299147297477691693786, 4.94833317957861687801544182154, 5.59202534097368005696866685373, 6.51227139341212340002414254320, 7.05326854889762029001362538216, 7.975217029001100362585484797536, 8.647265804045718476556355812988

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