Properties

Label 2-228e2-1.1-c1-0-25
Degree $2$
Conductor $51984$
Sign $1$
Analytic cond. $415.094$
Root an. cond. $20.3738$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 7-s + 3·11-s + 4·13-s + 3·17-s + 4·25-s + 6·29-s − 4·31-s − 3·35-s − 2·37-s − 6·41-s + 43-s − 3·47-s − 6·49-s + 12·53-s − 9·55-s + 6·59-s − 61-s − 12·65-s − 4·67-s − 6·71-s − 7·73-s + 3·77-s + 8·79-s + 12·83-s − 9·85-s + 12·89-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.377·7-s + 0.904·11-s + 1.10·13-s + 0.727·17-s + 4/5·25-s + 1.11·29-s − 0.718·31-s − 0.507·35-s − 0.328·37-s − 0.937·41-s + 0.152·43-s − 0.437·47-s − 6/7·49-s + 1.64·53-s − 1.21·55-s + 0.781·59-s − 0.128·61-s − 1.48·65-s − 0.488·67-s − 0.712·71-s − 0.819·73-s + 0.341·77-s + 0.900·79-s + 1.31·83-s − 0.976·85-s + 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51984\)    =    \(2^{4} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(415.094\)
Root analytic conductor: \(20.3738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 51984,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.186586965\)
\(L(\frac12)\) \(\approx\) \(2.186586965\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.52069919537353, −14.06036870188371, −13.47938618111398, −12.92017371056357, −12.23332794748519, −11.82418337098126, −11.56506835232669, −10.99444170964555, −10.42395361370351, −9.916723032273859, −9.082799743212163, −8.579318020473005, −8.336100555324519, −7.573712692572732, −7.247598908997903, −6.481595159971530, −6.062406338666524, −5.212564481269915, −4.667636823268672, −3.969854336520860, −3.582571885663239, −3.105748941402877, −1.993206817384209, −1.239024518098706, −0.5814224729591374, 0.5814224729591374, 1.239024518098706, 1.993206817384209, 3.105748941402877, 3.582571885663239, 3.969854336520860, 4.667636823268672, 5.212564481269915, 6.062406338666524, 6.481595159971530, 7.247598908997903, 7.573712692572732, 8.336100555324519, 8.579318020473005, 9.082799743212163, 9.916723032273859, 10.42395361370351, 10.99444170964555, 11.56506835232669, 11.82418337098126, 12.23332794748519, 12.92017371056357, 13.47938618111398, 14.06036870188371, 14.52069919537353

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