Properties

Label 2-63e2-1.1-c1-0-123
Degree $2$
Conductor $3969$
Sign $1$
Analytic cond. $31.6926$
Root an. cond. $5.62962$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.46·2-s + 4.05·4-s + 3.65·5-s + 5.05·8-s + 9.00·10-s + 0.406·11-s + 0.486·13-s + 4.32·16-s − 4.85·17-s + 1.97·19-s + 14.8·20-s + 22-s + 4.64·23-s + 8.38·25-s + 1.19·26-s + 7.64·29-s − 7.02·31-s + 0.539·32-s − 11.9·34-s + 2.32·37-s + 4.85·38-s + 18.4·40-s − 7.51·41-s − 2.32·43-s + 1.64·44-s + 11.4·46-s + 6.31·47-s + ⋯
L(s)  = 1  + 1.73·2-s + 2.02·4-s + 1.63·5-s + 1.78·8-s + 2.84·10-s + 0.122·11-s + 0.135·13-s + 1.08·16-s − 1.17·17-s + 0.452·19-s + 3.31·20-s + 0.213·22-s + 0.969·23-s + 1.67·25-s + 0.234·26-s + 1.42·29-s − 1.26·31-s + 0.0953·32-s − 2.04·34-s + 0.382·37-s + 0.787·38-s + 2.92·40-s − 1.17·41-s − 0.354·43-s + 0.248·44-s + 1.68·46-s + 0.921·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.6926\)
Root analytic conductor: \(5.62962\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.723005642\)
\(L(\frac12)\) \(\approx\) \(7.723005642\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - 2.46T + 2T^{2} \)
5 \( 1 - 3.65T + 5T^{2} \)
11 \( 1 - 0.406T + 11T^{2} \)
13 \( 1 - 0.486T + 13T^{2} \)
17 \( 1 + 4.85T + 17T^{2} \)
19 \( 1 - 1.97T + 19T^{2} \)
23 \( 1 - 4.64T + 23T^{2} \)
29 \( 1 - 7.64T + 29T^{2} \)
31 \( 1 + 7.02T + 31T^{2} \)
37 \( 1 - 2.32T + 37T^{2} \)
41 \( 1 + 7.51T + 41T^{2} \)
43 \( 1 + 2.32T + 43T^{2} \)
47 \( 1 - 6.31T + 47T^{2} \)
53 \( 1 + 3.56T + 53T^{2} \)
59 \( 1 + 6.11T + 59T^{2} \)
61 \( 1 - 8.02T + 61T^{2} \)
67 \( 1 - 3.60T + 67T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 + 1.97T + 73T^{2} \)
79 \( 1 - 8.16T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 - 9.48T + 97T^{2} \)
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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

−8.533354789291966931894408913405, −7.19223399074447779364407073663, −6.62752984795970189280873025494, −6.12651965665229332774319142878, −5.29128406163359026380324102533, −4.93271570063735789047803956246, −3.95997914762161076373903823865, −2.96101010513588786207001810354, −2.32564849613131091216071532632, −1.44306510287697189816581458684, 1.44306510287697189816581458684, 2.32564849613131091216071532632, 2.96101010513588786207001810354, 3.95997914762161076373903823865, 4.93271570063735789047803956246, 5.29128406163359026380324102533, 6.12651965665229332774319142878, 6.62752984795970189280873025494, 7.19223399074447779364407073663, 8.533354789291966931894408913405

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