Properties

Label 2-825-1.1-c5-0-54
Degree $2$
Conductor $825$
Sign $1$
Analytic cond. $132.316$
Root an. cond. $11.5028$
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  − 9.20·2-s − 9·3-s + 52.8·4-s + 82.8·6-s + 195.·7-s − 191.·8-s + 81·9-s + 121·11-s − 475.·12-s + 457.·13-s − 1.79e3·14-s + 74.5·16-s − 1.23e3·17-s − 745.·18-s + 2.14e3·19-s − 1.75e3·21-s − 1.11e3·22-s − 2.27e3·23-s + 1.72e3·24-s − 4.20e3·26-s − 729·27-s + 1.03e4·28-s + 6.56e3·29-s + 6.34e3·31-s + 5.44e3·32-s − 1.08e3·33-s + 1.13e4·34-s + ⋯
L(s)  = 1  − 1.62·2-s − 0.577·3-s + 1.65·4-s + 0.939·6-s + 1.50·7-s − 1.05·8-s + 0.333·9-s + 0.301·11-s − 0.952·12-s + 0.750·13-s − 2.45·14-s + 0.0727·16-s − 1.03·17-s − 0.542·18-s + 1.36·19-s − 0.869·21-s − 0.490·22-s − 0.895·23-s + 0.611·24-s − 1.22·26-s − 0.192·27-s + 2.48·28-s + 1.44·29-s + 1.18·31-s + 0.939·32-s − 0.174·33-s + 1.69·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(132.316\)
Root analytic conductor: \(11.5028\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.166267644\)
\(L(\frac12)\) \(\approx\) \(1.166267644\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
5 \( 1 \)
11 \( 1 - 121T \)
good2 \( 1 + 9.20T + 32T^{2} \)
7 \( 1 - 195.T + 1.68e4T^{2} \)
13 \( 1 - 457.T + 3.71e5T^{2} \)
17 \( 1 + 1.23e3T + 1.41e6T^{2} \)
19 \( 1 - 2.14e3T + 2.47e6T^{2} \)
23 \( 1 + 2.27e3T + 6.43e6T^{2} \)
29 \( 1 - 6.56e3T + 2.05e7T^{2} \)
31 \( 1 - 6.34e3T + 2.86e7T^{2} \)
37 \( 1 - 3.37e3T + 6.93e7T^{2} \)
41 \( 1 - 3.97e3T + 1.15e8T^{2} \)
43 \( 1 + 3.90e3T + 1.47e8T^{2} \)
47 \( 1 - 1.26e3T + 2.29e8T^{2} \)
53 \( 1 + 1.05e4T + 4.18e8T^{2} \)
59 \( 1 - 4.35e4T + 7.14e8T^{2} \)
61 \( 1 - 4.90e4T + 8.44e8T^{2} \)
67 \( 1 + 1.47e4T + 1.35e9T^{2} \)
71 \( 1 - 8.96e3T + 1.80e9T^{2} \)
73 \( 1 - 8.33e4T + 2.07e9T^{2} \)
79 \( 1 + 1.78e4T + 3.07e9T^{2} \)
83 \( 1 - 6.68e4T + 3.93e9T^{2} \)
89 \( 1 + 1.17e5T + 5.58e9T^{2} \)
97 \( 1 + 1.56e5T + 8.58e9T^{2} \)
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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

−9.511382385324016098033969369065, −8.381736500016067783340927629536, −8.184568931203580174164897188430, −7.10404322587860737261073491922, −6.32970673354078512783129919674, −5.13773937567835598397484787486, −4.18393439896763631205578399711, −2.40491554203558628746473248747, −1.36231460622469525076075752786, −0.74632116432464308477409735919, 0.74632116432464308477409735919, 1.36231460622469525076075752786, 2.40491554203558628746473248747, 4.18393439896763631205578399711, 5.13773937567835598397484787486, 6.32970673354078512783129919674, 7.10404322587860737261073491922, 8.184568931203580174164897188430, 8.381736500016067783340927629536, 9.511382385324016098033969369065

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