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Properties

Label 2-3420-19.11-c1-0-1
Degree $2$
Conductor $3420$
Sign $-0.0977 - 0.995i$
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 − 4·7-s + 3·11-s + (−3 − 5.19i)13-s + (1 − 1.73i)17-s + (3.5 + 2.59i)19-s + (2 + 3.46i)23-s + (−0.499 − 0.866i)25-s + (0.5 + 0.866i)29-s − 5·31-s + (2 − 3.46i)35-s − 4·37-s + (1 − 1.73i)41-s + (3 + 5.19i)47-s + 9·49-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s − 1.51·7-s + 0.904·11-s + (−0.832 − 1.44i)13-s + (0.242 − 0.420i)17-s + (0.802 + 0.596i)19-s + (0.417 + 0.722i)23-s + (−0.0999 − 0.173i)25-s + (0.0928 + 0.160i)29-s − 0.898·31-s + (0.338 − 0.585i)35-s − 0.657·37-s + (0.156 − 0.270i)41-s + (0.437 + 0.757i)47-s + 1.28·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.0977 - 0.995i$
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: \((2,\ 3420,\ (\ :1/2),\ -0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8689719260\)
\(L(\frac12)\) \(\approx\) \(0.8689719260\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 2.59i)T \)
good7 \( 1 + 4T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7 - 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.5 + 12.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (-8.5 - 14.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6 - 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.966891459658000588404092848485, −7.88049762224306117566402467395, −7.24937173782019793001168084143, −6.68645229352955628743311182619, −5.76357538241750172154811519651, −5.20924209378996784352518722786, −3.81358153883132156076803315015, −3.33588375185845154265011554385, −2.58118126271911375319498048089, −0.984050751173737153450723924121, 0.31317974222939298303837249091, 1.70956062175205331654767734095, 2.88007166650567570647748443265, 3.73002428415643947672550827682, 4.43717947030611179409087808746, 5.36841780246809574208281192990, 6.38226162138481808338497080014, 6.83641007431330357023185711493, 7.45990197158958493512580376178, 8.651901881445845856311804418725

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