Properties

Label 2-63e2-1.1-c1-0-128
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·2-s − 1.61·4-s + 2.28·5-s + 2.23·8-s − 1.41·10-s + 11-s + 4.57·13-s + 1.85·16-s − 3.16·17-s − 4.03·19-s − 3.70·20-s − 0.618·22-s − 7.23·23-s + 0.236·25-s − 2.82·26-s + 1.23·29-s − 5.99·31-s − 5.61·32-s + 1.95·34-s − 10.7·37-s + 2.49·38-s + 5.11·40-s − 5.78·41-s + 5.94·43-s − 1.61·44-s + 4.47·46-s − 3.03·47-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.809·4-s + 1.02·5-s + 0.790·8-s − 0.447·10-s + 0.301·11-s + 1.26·13-s + 0.463·16-s − 0.766·17-s − 0.925·19-s − 0.827·20-s − 0.131·22-s − 1.50·23-s + 0.0472·25-s − 0.554·26-s + 0.229·29-s − 1.07·31-s − 0.993·32-s + 0.335·34-s − 1.76·37-s + 0.404·38-s + 0.809·40-s − 0.903·41-s + 0.906·43-s − 0.243·44-s + 0.659·46-s − 0.442·47-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: \(1\)
Selberg data: \((2,\ 3969,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.618T + 2T^{2} \)
5 \( 1 - 2.28T + 5T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 - 4.57T + 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 + 4.03T + 19T^{2} \)
23 \( 1 + 7.23T + 23T^{2} \)
29 \( 1 - 1.23T + 29T^{2} \)
31 \( 1 + 5.99T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + 5.78T + 41T^{2} \)
43 \( 1 - 5.94T + 43T^{2} \)
47 \( 1 + 3.03T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 + 14.4T + 59T^{2} \)
61 \( 1 - 8.94T + 61T^{2} \)
67 \( 1 - 1.47T + 67T^{2} \)
71 \( 1 - 5.47T + 71T^{2} \)
73 \( 1 + 6.73T + 73T^{2} \)
79 \( 1 + 7.76T + 79T^{2} \)
83 \( 1 + 8.61T + 83T^{2} \)
89 \( 1 - 6.32T + 89T^{2} \)
97 \( 1 - 6.19T + 97T^{2} \)
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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

−8.484469838420441807326260117490, −7.45398032013452852640016531259, −6.48936059110705667759250902316, −5.92471521385459265558799499114, −5.18057375771130797176391201197, −4.17246377236454117374214944959, −3.62685269407757783154481104067, −2.12401430888258278763190651210, −1.47269916421327283812331377086, 0, 1.47269916421327283812331377086, 2.12401430888258278763190651210, 3.62685269407757783154481104067, 4.17246377236454117374214944959, 5.18057375771130797176391201197, 5.92471521385459265558799499114, 6.48936059110705667759250902316, 7.45398032013452852640016531259, 8.484469838420441807326260117490

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