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Properties

Label 2-63e2-3.2-c0-0-2
Degree $2$
Conductor $3969$
Sign $-1$
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  − 1.93i·2-s − 2.73·4-s + 3.34i·8-s + 0.517i·11-s + 3.73·16-s + 0.999·22-s − 1.41i·23-s + 25-s − 1.41i·29-s − 3.86i·32-s + 1.73·37-s − 1.73·43-s − 1.41i·44-s − 2.73·46-s − 1.93i·50-s + ⋯
L(s)  = 1  − 1.93i·2-s − 2.73·4-s + 3.34i·8-s + 0.517i·11-s + 3.73·16-s + 0.999·22-s − 1.41i·23-s + 25-s − 1.41i·29-s − 3.86i·32-s + 1.73·37-s − 1.73·43-s − 1.41i·44-s − 2.73·46-s − 1.93i·50-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3725, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9553796115\)
\(L(\frac12)\) \(\approx\) \(0.9553796115\)
\(L(1)\) 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 \( 1 \)
good2 \( 1 + 1.93iT - T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 - 0.517iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.73T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.73T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.517iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + 1.93iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.463311993714024538457640353503, −7.997797489445745322720673568909, −6.77882796565483789517463269972, −5.78296006982442827819035302821, −4.68668345871981838052957328624, −4.43821875281113136516768523565, −3.38336366409589532500310773252, −2.60679355840232943341171392509, −1.87254659880856011903497559680, −0.65498837616697276601980948779, 1.15120527358256937590115026030, 3.11028620019379152938681199562, 3.93528692086243315427971413839, 4.87394311557662320559463637253, 5.42421416350708734240397328349, 6.13133776019401321388510169051, 6.86786957467052075836182543472, 7.39149750615712762349894314834, 8.190905973288598584106637796437, 8.683054102841146517862818277031

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