Properties

Label 2-63e2-1.1-c1-0-92
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  + 2.37·2-s + 3.62·4-s + 1.22·5-s + 3.85·8-s + 2.91·10-s − 2.66·11-s + 3.62·13-s + 1.88·16-s + 6.73·17-s − 2.51·19-s + 4.45·20-s − 6.31·22-s + 7.99·23-s − 3.49·25-s + 8.59·26-s − 2.24·29-s + 10.2·31-s − 3.22·32-s + 15.9·34-s − 3.53·37-s − 5.96·38-s + 4.73·40-s + 1.86·41-s + 5.13·43-s − 9.64·44-s + 18.9·46-s − 2.14·47-s + ⋯
L(s)  = 1  + 1.67·2-s + 1.81·4-s + 0.549·5-s + 1.36·8-s + 0.921·10-s − 0.802·11-s + 1.00·13-s + 0.472·16-s + 1.63·17-s − 0.576·19-s + 0.995·20-s − 1.34·22-s + 1.66·23-s − 0.698·25-s + 1.68·26-s − 0.417·29-s + 1.83·31-s − 0.570·32-s + 2.73·34-s − 0.581·37-s − 0.967·38-s + 0.748·40-s + 0.291·41-s + 0.783·43-s − 1.45·44-s + 2.79·46-s − 0.313·47-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

\(L(1)\) \(\approx\) \(6.199385678\)
\(L(\frac12)\) \(\approx\) \(6.199385678\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.37T + 2T^{2} \)
5 \( 1 - 1.22T + 5T^{2} \)
11 \( 1 + 2.66T + 11T^{2} \)
13 \( 1 - 3.62T + 13T^{2} \)
17 \( 1 - 6.73T + 17T^{2} \)
19 \( 1 + 2.51T + 19T^{2} \)
23 \( 1 - 7.99T + 23T^{2} \)
29 \( 1 + 2.24T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 3.53T + 37T^{2} \)
41 \( 1 - 1.86T + 41T^{2} \)
43 \( 1 - 5.13T + 43T^{2} \)
47 \( 1 + 2.14T + 47T^{2} \)
53 \( 1 + 2.97T + 53T^{2} \)
59 \( 1 + 8.72T + 59T^{2} \)
61 \( 1 - 15.0T + 61T^{2} \)
67 \( 1 + 2.64T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + 7.29T + 73T^{2} \)
79 \( 1 + 0.313T + 79T^{2} \)
83 \( 1 - 7.69T + 83T^{2} \)
89 \( 1 + 7.19T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−8.264016938824481290228083346029, −7.52682632577267670645690314061, −6.61871455542520632773611231573, −6.01055280350625975904708096597, −5.40490235577959264150626994386, −4.82041202547137967204795739007, −3.84622118287558747075970123859, −3.13718450495830824514778001766, −2.40813954636209964861114108291, −1.21790938558758227191926983072, 1.21790938558758227191926983072, 2.40813954636209964861114108291, 3.13718450495830824514778001766, 3.84622118287558747075970123859, 4.82041202547137967204795739007, 5.40490235577959264150626994386, 6.01055280350625975904708096597, 6.61871455542520632773611231573, 7.52682632577267670645690314061, 8.264016938824481290228083346029

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