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Properties

Label 2-3420-5.4-c1-0-21
Degree $2$
Conductor $3420$
Sign $0.853 - 0.521i$
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  + (1.90 − 1.16i)5-s + 1.36i·7-s + 0.469·11-s − 1.03i·13-s + 5.65i·17-s + 19-s + 2.30i·23-s + (2.27 − 4.45i)25-s − 6.21·29-s + 9.98·31-s + (1.59 + 2.61i)35-s + 8.64i·37-s − 0.654·41-s − 8.26i·43-s + 8.39i·47-s + ⋯
L(s)  = 1  + (0.853 − 0.521i)5-s + 0.517i·7-s + 0.141·11-s − 0.287i·13-s + 1.37i·17-s + 0.229·19-s + 0.481i·23-s + (0.455 − 0.890i)25-s − 1.15·29-s + 1.79·31-s + (0.270 + 0.441i)35-s + 1.42i·37-s − 0.102·41-s − 1.25i·43-s + 1.22i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.853 - 0.521i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ 0.853 - 0.521i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.211366634\)
\(L(\frac12)\) \(\approx\) \(2.211366634\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.90 + 1.16i)T \)
19 \( 1 - T \)
good7 \( 1 - 1.36iT - 7T^{2} \)
11 \( 1 - 0.469T + 11T^{2} \)
13 \( 1 + 1.03iT - 13T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
23 \( 1 - 2.30iT - 23T^{2} \)
29 \( 1 + 6.21T + 29T^{2} \)
31 \( 1 - 9.98T + 31T^{2} \)
37 \( 1 - 8.64iT - 37T^{2} \)
41 \( 1 + 0.654T + 41T^{2} \)
43 \( 1 + 8.26iT - 43T^{2} \)
47 \( 1 - 8.39iT - 47T^{2} \)
53 \( 1 - 9.24iT - 53T^{2} \)
59 \( 1 - 5.62T + 59T^{2} \)
61 \( 1 - 9.06T + 61T^{2} \)
67 \( 1 + 5.62iT - 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 3.89iT - 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 - 3.61iT - 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 - 5.65iT - 97T^{2} \)
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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

−8.648722242441122714610481132297, −8.143971549821259273700509505818, −7.15475731697726412328370033105, −6.13609935298853175053281721620, −5.81238650983852422935328272987, −4.94145557350775695825327858419, −4.08509931367580948933795075960, −3.00106094784375377413225099158, −2.03495794383950791705374408286, −1.12495215174089926258797294693, 0.74939672425063656806327347479, 2.04072173307122686505422320972, 2.83823083447722751402261429518, 3.81179514680015976345816610097, 4.77729651842308097591500536375, 5.53870893288597973889036246907, 6.37763936875998507335825777795, 7.04714368381330824645423961861, 7.58033389741407798993659706844, 8.678353088499347135587150971621

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