Properties

Label 2-825-1.1-c5-0-93
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.33·2-s − 9·3-s − 20.8·4-s − 30.0·6-s − 103.·7-s − 176.·8-s + 81·9-s + 121·11-s + 187.·12-s + 357.·13-s − 345.·14-s + 78.7·16-s − 2.06e3·17-s + 270.·18-s + 1.32e3·19-s + 931.·21-s + 403.·22-s + 2.49e3·23-s + 1.58e3·24-s + 1.19e3·26-s − 729·27-s + 2.15e3·28-s + 1.65e3·29-s + 4.69e3·31-s + 5.90e3·32-s − 1.08e3·33-s − 6.90e3·34-s + ⋯
L(s)  = 1  + 0.589·2-s − 0.577·3-s − 0.651·4-s − 0.340·6-s − 0.798·7-s − 0.974·8-s + 0.333·9-s + 0.301·11-s + 0.376·12-s + 0.587·13-s − 0.470·14-s + 0.0769·16-s − 1.73·17-s + 0.196·18-s + 0.838·19-s + 0.460·21-s + 0.177·22-s + 0.984·23-s + 0.562·24-s + 0.346·26-s − 0.192·27-s + 0.520·28-s + 0.366·29-s + 0.876·31-s + 1.01·32-s − 0.174·33-s − 1.02·34-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: \(1\)
Selberg data: \((2,\ 825,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 3.33T + 32T^{2} \)
7 \( 1 + 103.T + 1.68e4T^{2} \)
13 \( 1 - 357.T + 3.71e5T^{2} \)
17 \( 1 + 2.06e3T + 1.41e6T^{2} \)
19 \( 1 - 1.32e3T + 2.47e6T^{2} \)
23 \( 1 - 2.49e3T + 6.43e6T^{2} \)
29 \( 1 - 1.65e3T + 2.05e7T^{2} \)
31 \( 1 - 4.69e3T + 2.86e7T^{2} \)
37 \( 1 + 5.21e3T + 6.93e7T^{2} \)
41 \( 1 - 1.35e4T + 1.15e8T^{2} \)
43 \( 1 + 1.07e4T + 1.47e8T^{2} \)
47 \( 1 - 2.92e3T + 2.29e8T^{2} \)
53 \( 1 - 2.15e4T + 4.18e8T^{2} \)
59 \( 1 + 4.54e3T + 7.14e8T^{2} \)
61 \( 1 - 3.64e3T + 8.44e8T^{2} \)
67 \( 1 - 5.54e4T + 1.35e9T^{2} \)
71 \( 1 - 4.49e3T + 1.80e9T^{2} \)
73 \( 1 + 3.35e4T + 2.07e9T^{2} \)
79 \( 1 + 7.91e3T + 3.07e9T^{2} \)
83 \( 1 - 2.05e4T + 3.93e9T^{2} \)
89 \( 1 + 1.11e5T + 5.58e9T^{2} \)
97 \( 1 - 5.82e4T + 8.58e9T^{2} \)
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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

−9.134245896021373828759355317574, −8.385091086892319959995081706095, −6.93542446036624537506754284436, −6.36916973724689421080356140584, −5.44060381845518383267413137523, −4.57175540869350877941919341571, −3.74713101007137511821016943450, −2.72596201164065504361810634082, −1.02613862856207966090544178669, 0, 1.02613862856207966090544178669, 2.72596201164065504361810634082, 3.74713101007137511821016943450, 4.57175540869350877941919341571, 5.44060381845518383267413137523, 6.36916973724689421080356140584, 6.93542446036624537506754284436, 8.385091086892319959995081706095, 9.134245896021373828759355317574

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