Properties

Label 2-825-1.1-c5-0-126
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  + 1.48·2-s + 9·3-s − 29.7·4-s + 13.3·6-s + 7.94·7-s − 91.9·8-s + 81·9-s − 121·11-s − 268.·12-s − 541.·13-s + 11.8·14-s + 816.·16-s + 1.07e3·17-s + 120.·18-s + 809.·19-s + 71.4·21-s − 180.·22-s − 614.·23-s − 827.·24-s − 805.·26-s + 729·27-s − 236.·28-s − 552.·29-s + 9.76e3·31-s + 4.15e3·32-s − 1.08e3·33-s + 1.59e3·34-s + ⋯
L(s)  = 1  + 0.263·2-s + 0.577·3-s − 0.930·4-s + 0.151·6-s + 0.0612·7-s − 0.507·8-s + 0.333·9-s − 0.301·11-s − 0.537·12-s − 0.888·13-s + 0.0161·14-s + 0.797·16-s + 0.898·17-s + 0.0876·18-s + 0.514·19-s + 0.0353·21-s − 0.0793·22-s − 0.242·23-s − 0.293·24-s − 0.233·26-s + 0.192·27-s − 0.0570·28-s − 0.121·29-s + 1.82·31-s + 0.717·32-s − 0.174·33-s + 0.236·34-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 \( 1 - 1.48T + 32T^{2} \)
7 \( 1 - 7.94T + 1.68e4T^{2} \)
13 \( 1 + 541.T + 3.71e5T^{2} \)
17 \( 1 - 1.07e3T + 1.41e6T^{2} \)
19 \( 1 - 809.T + 2.47e6T^{2} \)
23 \( 1 + 614.T + 6.43e6T^{2} \)
29 \( 1 + 552.T + 2.05e7T^{2} \)
31 \( 1 - 9.76e3T + 2.86e7T^{2} \)
37 \( 1 - 839.T + 6.93e7T^{2} \)
41 \( 1 + 1.12e4T + 1.15e8T^{2} \)
43 \( 1 + 9.56e3T + 1.47e8T^{2} \)
47 \( 1 - 2.03e4T + 2.29e8T^{2} \)
53 \( 1 + 2.16e4T + 4.18e8T^{2} \)
59 \( 1 + 8.64e3T + 7.14e8T^{2} \)
61 \( 1 + 3.27e4T + 8.44e8T^{2} \)
67 \( 1 - 4.40e4T + 1.35e9T^{2} \)
71 \( 1 + 2.33e4T + 1.80e9T^{2} \)
73 \( 1 + 7.91e4T + 2.07e9T^{2} \)
79 \( 1 - 3.32e4T + 3.07e9T^{2} \)
83 \( 1 - 1.13e5T + 3.93e9T^{2} \)
89 \( 1 + 1.98e4T + 5.58e9T^{2} \)
97 \( 1 - 8.57e3T + 8.58e9T^{2} \)
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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

−9.106945689662244708364545606649, −8.141086191232257951608904807460, −7.63280484306165743646850212502, −6.37619762155957936436907458956, −5.24647250311313720643563922437, −4.61195585779719093699416138108, −3.51570275816824800943090721176, −2.69524565772462769361503746119, −1.24579311250788875004589322948, 0, 1.24579311250788875004589322948, 2.69524565772462769361503746119, 3.51570275816824800943090721176, 4.61195585779719093699416138108, 5.24647250311313720643563922437, 6.37619762155957936436907458956, 7.63280484306165743646850212502, 8.141086191232257951608904807460, 9.106945689662244708364545606649

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