Properties

Label 2-15e2-1.1-c5-0-0
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $36.0863$
Root an. cond. $6.00719$
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  − 4.47·2-s − 11.9·4-s − 224.·7-s + 196.·8-s − 501.·11-s − 224.·13-s + 1.00e3·14-s − 496.·16-s − 1.66e3·17-s + 484·19-s + 2.24e3·22-s − 2.26e3·23-s + 1.00e3·26-s + 2.69e3·28-s − 5.52e3·29-s + 3.60e3·31-s − 4.07e3·32-s + 7.46e3·34-s − 7.40e3·37-s − 2.16e3·38-s + 1.10e4·41-s − 1.25e4·43-s + 6.02e3·44-s + 1.01e4·46-s − 9.54e3·47-s + 3.35e4·49-s + 2.69e3·52-s + ⋯
L(s)  = 1  − 0.790·2-s − 0.374·4-s − 1.73·7-s + 1.08·8-s − 1.25·11-s − 0.368·13-s + 1.36·14-s − 0.484·16-s − 1.39·17-s + 0.307·19-s + 0.988·22-s − 0.891·23-s + 0.291·26-s + 0.649·28-s − 1.21·29-s + 0.674·31-s − 0.704·32-s + 1.10·34-s − 0.889·37-s − 0.243·38-s + 1.02·41-s − 1.03·43-s + 0.469·44-s + 0.705·46-s − 0.630·47-s + 1.99·49-s + 0.138·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(36.0863\)
Root analytic conductor: \(6.00719\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.2160585139\)
\(L(\frac12)\) \(\approx\) \(0.2160585139\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 4.47T + 32T^{2} \)
7 \( 1 + 224.T + 1.68e4T^{2} \)
11 \( 1 + 501.T + 1.61e5T^{2} \)
13 \( 1 + 224.T + 3.71e5T^{2} \)
17 \( 1 + 1.66e3T + 1.41e6T^{2} \)
19 \( 1 - 484T + 2.47e6T^{2} \)
23 \( 1 + 2.26e3T + 6.43e6T^{2} \)
29 \( 1 + 5.52e3T + 2.05e7T^{2} \)
31 \( 1 - 3.60e3T + 2.86e7T^{2} \)
37 \( 1 + 7.40e3T + 6.93e7T^{2} \)
41 \( 1 - 1.10e4T + 1.15e8T^{2} \)
43 \( 1 + 1.25e4T + 1.47e8T^{2} \)
47 \( 1 + 9.54e3T + 2.29e8T^{2} \)
53 \( 1 + 4.77e3T + 4.18e8T^{2} \)
59 \( 1 - 5.52e3T + 7.14e8T^{2} \)
61 \( 1 - 2.13e4T + 8.44e8T^{2} \)
67 \( 1 - 3.45e4T + 1.35e9T^{2} \)
71 \( 1 - 3.31e4T + 1.80e9T^{2} \)
73 \( 1 - 2.69e3T + 2.07e9T^{2} \)
79 \( 1 - 9.96e4T + 3.07e9T^{2} \)
83 \( 1 - 5.71e4T + 3.93e9T^{2} \)
89 \( 1 + 1.20e5T + 5.58e9T^{2} \)
97 \( 1 + 6.42e4T + 8.58e9T^{2} \)
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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

−11.04628532255598988332633466799, −10.05725409811950433603317381405, −9.573931262540467923243932479084, −8.564728362099047094053601344812, −7.50573997781794297036214276731, −6.46657420796636436536923048516, −5.12969983867077137146956585999, −3.74542760636028952782164098529, −2.33386683535418631157816049537, −0.29539854319508773744557523772, 0.29539854319508773744557523772, 2.33386683535418631157816049537, 3.74542760636028952782164098529, 5.12969983867077137146956585999, 6.46657420796636436536923048516, 7.50573997781794297036214276731, 8.564728362099047094053601344812, 9.573931262540467923243932479084, 10.05725409811950433603317381405, 11.04628532255598988332633466799

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