Properties

Label 2-63e2-3.2-c0-0-3
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  + 0.517i·2-s + 0.732·4-s + 0.896i·8-s − 1.93i·11-s + 0.267·16-s + 0.999·22-s − 1.41i·23-s + 25-s − 1.41i·29-s + 1.03i·32-s − 1.73·37-s + 1.73·43-s − 1.41i·44-s + 0.732·46-s + 0.517i·50-s + ⋯
L(s)  = 1  + 0.517i·2-s + 0.732·4-s + 0.896i·8-s − 1.93i·11-s + 0.267·16-s + 0.999·22-s − 1.41i·23-s + 25-s − 1.41i·29-s + 1.03i·32-s − 1.73·37-s + 1.73·43-s − 1.41i·44-s + 0.732·46-s + 0.517i·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

\(L(\frac{1}{2})\) \(\approx\) \(1.616730671\)
\(L(\frac12)\) \(\approx\) \(1.616730671\)
\(L(1)\) 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 \( 1 - 0.517iT - T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + 1.93iT - T^{2} \)
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 \( 1 - 1.93iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - 0.517iT - T^{2} \)
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

−8.574668238904785647650521647541, −7.901876518305277355443048978798, −7.12244528102868276957012557363, −6.30889748297332492022257215931, −5.92555473152344411233792404851, −5.12433118170940068414114745194, −4.04203104024629401648405164716, −3.03253884671891232351926797042, −2.40539446366987347553481715892, −0.951767437125475136559348898996, 1.44785938348991474911451193446, 2.08183176202678648937278717595, 3.09662523196564915871590242491, 3.88407554631482228822492648406, 4.86811669000795365775407743210, 5.55704033591448449378196093489, 6.77383929399435320026884319362, 7.05538804049269038003967621563, 7.68458949971094675799795762696, 8.764944194589420876815094318355

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