Properties

Label 2-39e2-1.1-c3-0-24
Degree $2$
Conductor $1521$
Sign $1$
Analytic cond. $89.7419$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.28·2-s − 6.34·4-s − 12.4·5-s − 9.00·7-s + 18.4·8-s + 16.0·10-s + 51.0·11-s + 11.5·14-s + 27.0·16-s + 6.86·17-s − 83.1·19-s + 78.9·20-s − 65.7·22-s − 187.·23-s + 29.9·25-s + 57.1·28-s + 223.·29-s + 57.3·31-s − 182.·32-s − 8.82·34-s + 112.·35-s − 156.·37-s + 106.·38-s − 229.·40-s − 222.·41-s + 347.·43-s − 324.·44-s + ⋯
L(s)  = 1  − 0.454·2-s − 0.793·4-s − 1.11·5-s − 0.486·7-s + 0.815·8-s + 0.506·10-s + 1.40·11-s + 0.221·14-s + 0.421·16-s + 0.0978·17-s − 1.00·19-s + 0.882·20-s − 0.636·22-s − 1.70·23-s + 0.239·25-s + 0.385·28-s + 1.43·29-s + 0.332·31-s − 1.00·32-s − 0.0445·34-s + 0.541·35-s − 0.694·37-s + 0.456·38-s − 0.908·40-s − 0.846·41-s + 1.23·43-s − 1.11·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(89.7419\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5512842836\)
\(L(\frac12)\) \(\approx\) \(0.5512842836\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + 1.28T + 8T^{2} \)
5 \( 1 + 12.4T + 125T^{2} \)
7 \( 1 + 9.00T + 343T^{2} \)
11 \( 1 - 51.0T + 1.33e3T^{2} \)
17 \( 1 - 6.86T + 4.91e3T^{2} \)
19 \( 1 + 83.1T + 6.85e3T^{2} \)
23 \( 1 + 187.T + 1.21e4T^{2} \)
29 \( 1 - 223.T + 2.43e4T^{2} \)
31 \( 1 - 57.3T + 2.97e4T^{2} \)
37 \( 1 + 156.T + 5.06e4T^{2} \)
41 \( 1 + 222.T + 6.89e4T^{2} \)
43 \( 1 - 347.T + 7.95e4T^{2} \)
47 \( 1 - 45.0T + 1.03e5T^{2} \)
53 \( 1 + 473.T + 1.48e5T^{2} \)
59 \( 1 + 615.T + 2.05e5T^{2} \)
61 \( 1 - 195.T + 2.26e5T^{2} \)
67 \( 1 + 355.T + 3.00e5T^{2} \)
71 \( 1 + 763.T + 3.57e5T^{2} \)
73 \( 1 - 331.T + 3.89e5T^{2} \)
79 \( 1 + 207.T + 4.93e5T^{2} \)
83 \( 1 - 251.T + 5.71e5T^{2} \)
89 \( 1 + 719.T + 7.04e5T^{2} \)
97 \( 1 + 1.55e3T + 9.12e5T^{2} \)
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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

−8.995588553781960468542442085918, −8.359053856290060037077512355685, −7.75705661900947438433322886143, −6.74029107802550855834069786594, −5.99941459147836720610803401659, −4.54278559292819085282586062627, −4.12985253564789038314076460013, −3.28720847252529450340796642212, −1.62215405411838707900674976178, −0.40216176888026715276851624571, 0.40216176888026715276851624571, 1.62215405411838707900674976178, 3.28720847252529450340796642212, 4.12985253564789038314076460013, 4.54278559292819085282586062627, 5.99941459147836720610803401659, 6.74029107802550855834069786594, 7.75705661900947438433322886143, 8.359053856290060037077512355685, 8.995588553781960468542442085918

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