Properties

Label 2-21e2-1.1-c5-0-29
Degree $2$
Conductor $441$
Sign $1$
Analytic cond. $70.7292$
Root an. cond. $8.41006$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.81·2-s + 29.0·4-s − 74.2·5-s − 22.8·8-s − 580.·10-s + 424.·11-s + 252.·13-s − 1.10e3·16-s − 1.10e3·17-s + 6.47·19-s − 2.15e3·20-s + 3.31e3·22-s + 3.61e3·23-s + 2.39e3·25-s + 1.97e3·26-s + 5.00e3·29-s − 2.82e3·31-s − 7.93e3·32-s − 8.63e3·34-s − 2.04e3·37-s + 50.5·38-s + 1.69e3·40-s + 9.39e3·41-s + 1.03e4·43-s + 1.23e4·44-s + 2.82e4·46-s + 1.70e4·47-s + ⋯
L(s)  = 1  + 1.38·2-s + 0.908·4-s − 1.32·5-s − 0.126·8-s − 1.83·10-s + 1.05·11-s + 0.413·13-s − 1.08·16-s − 0.926·17-s + 0.00411·19-s − 1.20·20-s + 1.46·22-s + 1.42·23-s + 0.765·25-s + 0.571·26-s + 1.10·29-s − 0.527·31-s − 1.36·32-s − 1.28·34-s − 0.245·37-s + 0.00568·38-s + 0.167·40-s + 0.872·41-s + 0.851·43-s + 0.960·44-s + 1.96·46-s + 1.12·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(70.7292\)
Root analytic conductor: \(8.41006\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.373131421\)
\(L(\frac12)\) \(\approx\) \(3.373131421\)
\(L(\frac{7}{2})\) 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 \( 1 - 7.81T + 32T^{2} \)
5 \( 1 + 74.2T + 3.12e3T^{2} \)
11 \( 1 - 424.T + 1.61e5T^{2} \)
13 \( 1 - 252.T + 3.71e5T^{2} \)
17 \( 1 + 1.10e3T + 1.41e6T^{2} \)
19 \( 1 - 6.47T + 2.47e6T^{2} \)
23 \( 1 - 3.61e3T + 6.43e6T^{2} \)
29 \( 1 - 5.00e3T + 2.05e7T^{2} \)
31 \( 1 + 2.82e3T + 2.86e7T^{2} \)
37 \( 1 + 2.04e3T + 6.93e7T^{2} \)
41 \( 1 - 9.39e3T + 1.15e8T^{2} \)
43 \( 1 - 1.03e4T + 1.47e8T^{2} \)
47 \( 1 - 1.70e4T + 2.29e8T^{2} \)
53 \( 1 - 3.95e4T + 4.18e8T^{2} \)
59 \( 1 - 3.39e4T + 7.14e8T^{2} \)
61 \( 1 + 2.82e4T + 8.44e8T^{2} \)
67 \( 1 - 5.61e4T + 1.35e9T^{2} \)
71 \( 1 - 1.55e4T + 1.80e9T^{2} \)
73 \( 1 - 7.82e4T + 2.07e9T^{2} \)
79 \( 1 + 4.53e4T + 3.07e9T^{2} \)
83 \( 1 - 1.38e3T + 3.93e9T^{2} \)
89 \( 1 + 6.88e4T + 5.58e9T^{2} \)
97 \( 1 + 1.08e5T + 8.58e9T^{2} \)
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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

−10.87098675570234083697267481314, −9.199660406517875437792664659970, −8.518127947859231462021282253720, −7.18442689324614687162457197183, −6.53620672013784157337492297482, −5.31960083440640789517428680599, −4.21662650325741952844136628595, −3.82129815268477657546611273202, −2.63659823885883562474776045585, −0.77639318743304759421234666333, 0.77639318743304759421234666333, 2.63659823885883562474776045585, 3.82129815268477657546611273202, 4.21662650325741952844136628595, 5.31960083440640789517428680599, 6.53620672013784157337492297482, 7.18442689324614687162457197183, 8.518127947859231462021282253720, 9.199660406517875437792664659970, 10.87098675570234083697267481314

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