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Properties

Label 2-3420-95.18-c1-0-12
Degree $2$
Conductor $3420$
Sign $0.480 - 0.877i$
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  + (−0.311 − 2.21i)5-s + (0.311 − 0.311i)7-s − 2.90·11-s + (−2.84 + 2.84i)13-s + (2.52 − 2.52i)17-s + (−4.34 − 0.377i)19-s + (4.11 + 4.11i)23-s + (−4.80 + 1.37i)25-s − 2.99·29-s + 0.930i·31-s + (−0.785 − 0.592i)35-s + (8.11 + 8.11i)37-s + 2.06i·41-s + (2.11 + 2.11i)43-s + (2.73 − 2.73i)47-s + ⋯
L(s)  = 1  + (−0.139 − 0.990i)5-s + (0.117 − 0.117i)7-s − 0.875·11-s + (−0.789 + 0.789i)13-s + (0.612 − 0.612i)17-s + (−0.996 − 0.0866i)19-s + (0.858 + 0.858i)23-s + (−0.961 + 0.275i)25-s − 0.555·29-s + 0.167i·31-s + (−0.132 − 0.100i)35-s + (1.33 + 1.33i)37-s + 0.321i·41-s + (0.322 + 0.322i)43-s + (0.399 − 0.399i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.480 - 0.877i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (2773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ 0.480 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.048559223\)
\(L(\frac12)\) \(\approx\) \(1.048559223\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.311 + 2.21i)T \)
19 \( 1 + (4.34 + 0.377i)T \)
good7 \( 1 + (-0.311 + 0.311i)T - 7iT^{2} \)
11 \( 1 + 2.90T + 11T^{2} \)
13 \( 1 + (2.84 - 2.84i)T - 13iT^{2} \)
17 \( 1 + (-2.52 + 2.52i)T - 17iT^{2} \)
23 \( 1 + (-4.11 - 4.11i)T + 23iT^{2} \)
29 \( 1 + 2.99T + 29T^{2} \)
31 \( 1 - 0.930iT - 31T^{2} \)
37 \( 1 + (-8.11 - 8.11i)T + 37iT^{2} \)
41 \( 1 - 2.06iT - 41T^{2} \)
43 \( 1 + (-2.11 - 2.11i)T + 43iT^{2} \)
47 \( 1 + (-2.73 + 2.73i)T - 47iT^{2} \)
53 \( 1 + (-0.565 + 0.565i)T - 53iT^{2} \)
59 \( 1 + 9.32T + 59T^{2} \)
61 \( 1 - 3.52T + 61T^{2} \)
67 \( 1 + (0.144 + 0.144i)T + 67iT^{2} \)
71 \( 1 - 9.61iT - 71T^{2} \)
73 \( 1 + (4.09 + 4.09i)T + 73iT^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + (9.21 + 9.21i)T + 83iT^{2} \)
89 \( 1 - 7.55T + 89T^{2} \)
97 \( 1 + (-12.0 - 12.0i)T + 97iT^{2} \)
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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

−8.806195032962800969448899460073, −7.83244520278290139796667753524, −7.52262304193458521572458529828, −6.48469119568802118792075721810, −5.54591218495711886330353358405, −4.84213262110216670225527808881, −4.34166529617252813377831731950, −3.15180575348505379933245736342, −2.15114569775001346532301876890, −1.01582141194263162409773946991, 0.35476673745841059718191239704, 2.15111624341941124957786788663, 2.75758332272760331815542448300, 3.68264945987434855869657556833, 4.63127655097907442633547405364, 5.57603509330544158488861141984, 6.16551559833566174493105310360, 7.13665675457080263655415902705, 7.69290681067572368136197057521, 8.281625184587283180703528581685

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