Properties

Label 2-825-1.1-c5-0-21
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  − 11.1·2-s − 9·3-s + 91.3·4-s + 99.9·6-s − 76.1·7-s − 659.·8-s + 81·9-s − 121·11-s − 822.·12-s + 470.·13-s + 845.·14-s + 4.39e3·16-s + 494.·17-s − 899.·18-s − 44.8·19-s + 685.·21-s + 1.34e3·22-s − 4.24e3·23-s + 5.93e3·24-s − 5.22e3·26-s − 729·27-s − 6.95e3·28-s − 786.·29-s + 5.59e3·31-s − 2.77e4·32-s + 1.08e3·33-s − 5.49e3·34-s + ⋯
L(s)  = 1  − 1.96·2-s − 0.577·3-s + 2.85·4-s + 1.13·6-s − 0.587·7-s − 3.64·8-s + 0.333·9-s − 0.301·11-s − 1.64·12-s + 0.772·13-s + 1.15·14-s + 4.29·16-s + 0.415·17-s − 0.654·18-s − 0.0284·19-s + 0.339·21-s + 0.591·22-s − 1.67·23-s + 2.10·24-s − 1.51·26-s − 0.192·27-s − 1.67·28-s − 0.173·29-s + 1.04·31-s − 4.79·32-s + 0.174·33-s − 0.815·34-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

\(L(3)\) \(\approx\) \(0.4196683800\)
\(L(\frac12)\) \(\approx\) \(0.4196683800\)
\(L(\frac{7}{2})\) 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 \( 1 + 11.1T + 32T^{2} \)
7 \( 1 + 76.1T + 1.68e4T^{2} \)
13 \( 1 - 470.T + 3.71e5T^{2} \)
17 \( 1 - 494.T + 1.41e6T^{2} \)
19 \( 1 + 44.8T + 2.47e6T^{2} \)
23 \( 1 + 4.24e3T + 6.43e6T^{2} \)
29 \( 1 + 786.T + 2.05e7T^{2} \)
31 \( 1 - 5.59e3T + 2.86e7T^{2} \)
37 \( 1 + 6.45e3T + 6.93e7T^{2} \)
41 \( 1 - 7.70e3T + 1.15e8T^{2} \)
43 \( 1 - 2.42e3T + 1.47e8T^{2} \)
47 \( 1 + 1.88e4T + 2.29e8T^{2} \)
53 \( 1 - 3.44e4T + 4.18e8T^{2} \)
59 \( 1 - 2.79e4T + 7.14e8T^{2} \)
61 \( 1 - 1.46e4T + 8.44e8T^{2} \)
67 \( 1 - 4.72e4T + 1.35e9T^{2} \)
71 \( 1 + 3.95e4T + 1.80e9T^{2} \)
73 \( 1 + 2.69e4T + 2.07e9T^{2} \)
79 \( 1 + 1.22e4T + 3.07e9T^{2} \)
83 \( 1 - 7.03e4T + 3.93e9T^{2} \)
89 \( 1 + 1.43e5T + 5.58e9T^{2} \)
97 \( 1 + 9.13e4T + 8.58e9T^{2} \)
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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

−9.693669935480029592133804429939, −8.576149942173421431191704640140, −8.018328065176613344384918697081, −7.04075956961536781589502968148, −6.31236626393907706976119303262, −5.61541854805478318471586673998, −3.72032255560524563512065364288, −2.52040986420128791001428398857, −1.41942410520883605914599261814, −0.42853353168514640284742684831, 0.42853353168514640284742684831, 1.41942410520883605914599261814, 2.52040986420128791001428398857, 3.72032255560524563512065364288, 5.61541854805478318471586673998, 6.31236626393907706976119303262, 7.04075956961536781589502968148, 8.018328065176613344384918697081, 8.576149942173421431191704640140, 9.693669935480029592133804429939

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