Properties

Label 2-825-1.1-c5-0-82
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  − 0.886·2-s + 9·3-s − 31.2·4-s − 7.97·6-s + 224.·7-s + 56.0·8-s + 81·9-s + 121·11-s − 280.·12-s + 1.06e3·13-s − 199.·14-s + 949.·16-s − 1.34e3·17-s − 71.8·18-s + 691.·19-s + 2.02e3·21-s − 107.·22-s + 3.39e3·23-s + 504.·24-s − 944.·26-s + 729·27-s − 7.01e3·28-s + 8.60e3·29-s − 320.·31-s − 2.63e3·32-s + 1.08e3·33-s + 1.19e3·34-s + ⋯
L(s)  = 1  − 0.156·2-s + 0.577·3-s − 0.975·4-s − 0.0904·6-s + 1.73·7-s + 0.309·8-s + 0.333·9-s + 0.301·11-s − 0.563·12-s + 1.74·13-s − 0.271·14-s + 0.926·16-s − 1.13·17-s − 0.0522·18-s + 0.439·19-s + 1.00·21-s − 0.0472·22-s + 1.33·23-s + 0.178·24-s − 0.274·26-s + 0.192·27-s − 1.69·28-s + 1.89·29-s − 0.0599·31-s − 0.454·32-s + 0.174·33-s + 0.177·34-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

\(L(3)\) \(\approx\) \(3.299564746\)
\(L(\frac12)\) \(\approx\) \(3.299564746\)
\(L(\frac{7}{2})\) 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 \( 1 + 0.886T + 32T^{2} \)
7 \( 1 - 224.T + 1.68e4T^{2} \)
13 \( 1 - 1.06e3T + 3.71e5T^{2} \)
17 \( 1 + 1.34e3T + 1.41e6T^{2} \)
19 \( 1 - 691.T + 2.47e6T^{2} \)
23 \( 1 - 3.39e3T + 6.43e6T^{2} \)
29 \( 1 - 8.60e3T + 2.05e7T^{2} \)
31 \( 1 + 320.T + 2.86e7T^{2} \)
37 \( 1 + 1.90e3T + 6.93e7T^{2} \)
41 \( 1 - 5.82e3T + 1.15e8T^{2} \)
43 \( 1 - 1.42e3T + 1.47e8T^{2} \)
47 \( 1 + 6.45e3T + 2.29e8T^{2} \)
53 \( 1 - 9.10e3T + 4.18e8T^{2} \)
59 \( 1 - 1.02e4T + 7.14e8T^{2} \)
61 \( 1 + 3.42e4T + 8.44e8T^{2} \)
67 \( 1 + 6.84e4T + 1.35e9T^{2} \)
71 \( 1 - 1.88e3T + 1.80e9T^{2} \)
73 \( 1 + 8.77e4T + 2.07e9T^{2} \)
79 \( 1 + 5.24e4T + 3.07e9T^{2} \)
83 \( 1 - 5.86e4T + 3.93e9T^{2} \)
89 \( 1 + 1.13e5T + 5.58e9T^{2} \)
97 \( 1 + 1.15e5T + 8.58e9T^{2} \)
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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

−9.023379865764342962666043178096, −8.702560491630807047413330712705, −8.129720827124642268803093278394, −7.12625554045203538088999916813, −5.86553830453701709960775299520, −4.71145252455652317400083014598, −4.29514817200029544740577060906, −3.07712473589331097144553039461, −1.56789029136306637902983738139, −0.952802378600297660443546801841, 0.952802378600297660443546801841, 1.56789029136306637902983738139, 3.07712473589331097144553039461, 4.29514817200029544740577060906, 4.71145252455652317400083014598, 5.86553830453701709960775299520, 7.12625554045203538088999916813, 8.129720827124642268803093278394, 8.702560491630807047413330712705, 9.023379865764342962666043178096

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