Properties

Label 2-63e2-1.1-c1-0-12
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.84·2-s + 1.39·4-s − 1.33·5-s + 1.12·8-s + 2.45·10-s + 1.51·11-s − 5.17·13-s − 4.84·16-s + 1.54·17-s − 2.50·19-s − 1.85·20-s − 2.78·22-s − 7.36·23-s − 3.21·25-s + 9.53·26-s + 0.0619·29-s + 3.84·31-s + 6.68·32-s − 2.85·34-s + 0.563·37-s + 4.61·38-s − 1.49·40-s + 9.02·41-s − 10.1·43-s + 2.10·44-s + 13.5·46-s + 9.51·47-s + ⋯
L(s)  = 1  − 1.30·2-s + 0.695·4-s − 0.596·5-s + 0.396·8-s + 0.777·10-s + 0.456·11-s − 1.43·13-s − 1.21·16-s + 0.375·17-s − 0.574·19-s − 0.414·20-s − 0.593·22-s − 1.53·23-s − 0.643·25-s + 1.86·26-s + 0.0115·29-s + 0.691·31-s + 1.18·32-s − 0.489·34-s + 0.0925·37-s + 0.747·38-s − 0.236·40-s + 1.40·41-s − 1.55·43-s + 0.317·44-s + 1.99·46-s + 1.38·47-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4515182849\)
\(L(\frac12)\) \(\approx\) \(0.4515182849\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.84T + 2T^{2} \)
5 \( 1 + 1.33T + 5T^{2} \)
11 \( 1 - 1.51T + 11T^{2} \)
13 \( 1 + 5.17T + 13T^{2} \)
17 \( 1 - 1.54T + 17T^{2} \)
19 \( 1 + 2.50T + 19T^{2} \)
23 \( 1 + 7.36T + 23T^{2} \)
29 \( 1 - 0.0619T + 29T^{2} \)
31 \( 1 - 3.84T + 31T^{2} \)
37 \( 1 - 0.563T + 37T^{2} \)
41 \( 1 - 9.02T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 9.51T + 47T^{2} \)
53 \( 1 + 1.51T + 53T^{2} \)
59 \( 1 - 8.44T + 59T^{2} \)
61 \( 1 + 3.23T + 61T^{2} \)
67 \( 1 - 6.93T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + 2.75T + 73T^{2} \)
79 \( 1 + 5.91T + 79T^{2} \)
83 \( 1 - 5.60T + 83T^{2} \)
89 \( 1 - 1.40T + 89T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.369922751788889287577859047216, −7.86962357555483632470672053743, −7.31860774710482593266968344339, −6.56272636609670014998051100248, −5.58642838442296380854000143443, −4.49153757906778863210234375846, −3.98537413264355085917955596625, −2.63155264406959070931373459786, −1.75990328610050205009647212101, −0.46395579872735933603755292742, 0.46395579872735933603755292742, 1.75990328610050205009647212101, 2.63155264406959070931373459786, 3.98537413264355085917955596625, 4.49153757906778863210234375846, 5.58642838442296380854000143443, 6.56272636609670014998051100248, 7.31860774710482593266968344339, 7.86962357555483632470672053743, 8.369922751788889287577859047216

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