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Properties

Label 2-63e2-441.223-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.935 - 0.352i$
Analytic cond. $1.98078$
Root an. cond. $1.40740$
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  + (0.988 + 0.149i)4-s + (−0.0747 + 0.997i)7-s + (1.09 − 1.61i)13-s + (0.955 + 0.294i)16-s + 0.867i·19-s + (0.826 − 0.563i)25-s + (−0.222 + 0.974i)28-s + (−1.35 + 0.781i)31-s + (−0.777 − 0.974i)37-s + (1.72 + 0.531i)43-s + (−0.988 − 0.149i)49-s + (1.32 − 1.42i)52-s + (−0.233 − 1.54i)61-s + (0.900 + 0.433i)64-s + (0.222 + 0.385i)67-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)4-s + (−0.0747 + 0.997i)7-s + (1.09 − 1.61i)13-s + (0.955 + 0.294i)16-s + 0.867i·19-s + (0.826 − 0.563i)25-s + (−0.222 + 0.974i)28-s + (−1.35 + 0.781i)31-s + (−0.777 − 0.974i)37-s + (1.72 + 0.531i)43-s + (−0.988 − 0.149i)49-s + (1.32 − 1.42i)52-s + (−0.233 − 1.54i)61-s + (0.900 + 0.433i)64-s + (0.222 + 0.385i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.935 - 0.352i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.935 - 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.768312371\)
\(L(\frac12)\) \(\approx\) \(1.768312371\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.0747 - 0.997i)T \)
good2 \( 1 + (-0.988 - 0.149i)T^{2} \)
5 \( 1 + (-0.826 + 0.563i)T^{2} \)
11 \( 1 + (-0.988 - 0.149i)T^{2} \)
13 \( 1 + (-1.09 + 1.61i)T + (-0.365 - 0.930i)T^{2} \)
17 \( 1 + (0.222 - 0.974i)T^{2} \)
19 \( 1 - 0.867iT - T^{2} \)
23 \( 1 + (0.955 + 0.294i)T^{2} \)
29 \( 1 + (0.955 - 0.294i)T^{2} \)
31 \( 1 + (1.35 - 0.781i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \)
41 \( 1 + (-0.826 + 0.563i)T^{2} \)
43 \( 1 + (-1.72 - 0.531i)T + (0.826 + 0.563i)T^{2} \)
47 \( 1 + (0.988 + 0.149i)T^{2} \)
53 \( 1 + (-0.222 - 0.974i)T^{2} \)
59 \( 1 + (-0.0747 + 0.997i)T^{2} \)
61 \( 1 + (0.233 + 1.54i)T + (-0.955 + 0.294i)T^{2} \)
67 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.365 + 0.930i)T^{2} \)
89 \( 1 + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + (-0.751 - 0.433i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.508810524210886115767615439457, −8.020920013530908977626910385309, −7.23797928267661180695620557262, −6.35766493440611612363593250116, −5.73639671749857204200110737649, −5.29276650057755779506701213037, −3.81550918545946104528016532302, −3.14160826044630963266910648498, −2.36871427675077047318207348448, −1.30059312510962713518133035041, 1.18330867687087268368152812312, 2.01682643978327091991288589201, 3.16985879639529062910758589104, 3.94284549942206946518008590503, 4.73966770472568395017416095579, 5.85152709858771361050942182014, 6.49416522397636571114364476463, 7.17409269601603193317753555265, 7.50602976837158335000508086861, 8.733166179662426468716804327965

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