Properties

Label 2-825-1.1-c5-0-103
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  + 2.92·2-s − 9·3-s − 23.4·4-s − 26.2·6-s + 85.0·7-s − 162.·8-s + 81·9-s − 121·11-s + 211.·12-s − 724.·13-s + 248.·14-s + 277.·16-s + 2.09e3·17-s + 236.·18-s − 6.40·19-s − 765.·21-s − 353.·22-s − 1.56e3·23-s + 1.45e3·24-s − 2.11e3·26-s − 729·27-s − 1.99e3·28-s − 5.14e3·29-s − 1.03e3·31-s + 5.99e3·32-s + 1.08e3·33-s + 6.13e3·34-s + ⋯
L(s)  = 1  + 0.516·2-s − 0.577·3-s − 0.733·4-s − 0.298·6-s + 0.655·7-s − 0.895·8-s + 0.333·9-s − 0.301·11-s + 0.423·12-s − 1.18·13-s + 0.338·14-s + 0.270·16-s + 1.76·17-s + 0.172·18-s − 0.00406·19-s − 0.378·21-s − 0.155·22-s − 0.618·23-s + 0.516·24-s − 0.613·26-s − 0.192·27-s − 0.480·28-s − 1.13·29-s − 0.192·31-s + 1.03·32-s + 0.174·33-s + 0.909·34-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: $\chi_{825} (1, \cdot )$
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 \( 1 - 2.92T + 32T^{2} \)
7 \( 1 - 85.0T + 1.68e4T^{2} \)
13 \( 1 + 724.T + 3.71e5T^{2} \)
17 \( 1 - 2.09e3T + 1.41e6T^{2} \)
19 \( 1 + 6.40T + 2.47e6T^{2} \)
23 \( 1 + 1.56e3T + 6.43e6T^{2} \)
29 \( 1 + 5.14e3T + 2.05e7T^{2} \)
31 \( 1 + 1.03e3T + 2.86e7T^{2} \)
37 \( 1 - 1.26e4T + 6.93e7T^{2} \)
41 \( 1 - 1.38e4T + 1.15e8T^{2} \)
43 \( 1 - 1.70e4T + 1.47e8T^{2} \)
47 \( 1 + 8.07e3T + 2.29e8T^{2} \)
53 \( 1 - 2.21e4T + 4.18e8T^{2} \)
59 \( 1 + 1.68e4T + 7.14e8T^{2} \)
61 \( 1 + 3.43e4T + 8.44e8T^{2} \)
67 \( 1 - 3.73e4T + 1.35e9T^{2} \)
71 \( 1 + 5.66e4T + 1.80e9T^{2} \)
73 \( 1 - 2.77e4T + 2.07e9T^{2} \)
79 \( 1 + 1.27e4T + 3.07e9T^{2} \)
83 \( 1 - 6.92e4T + 3.93e9T^{2} \)
89 \( 1 - 5.90e4T + 5.58e9T^{2} \)
97 \( 1 + 1.04e5T + 8.58e9T^{2} \)
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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

−9.294962060099826244160204489984, −7.914330285723199381888454815096, −7.55712335077672776302365328732, −6.02916301198691018755960297134, −5.42747433102649607331012164044, −4.68436205643931953604344196178, −3.81257051325707501448787147813, −2.56557075570168019424297531847, −1.10079465515973933975113574052, 0, 1.10079465515973933975113574052, 2.56557075570168019424297531847, 3.81257051325707501448787147813, 4.68436205643931953604344196178, 5.42747433102649607331012164044, 6.02916301198691018755960297134, 7.55712335077672776302365328732, 7.914330285723199381888454815096, 9.294962060099826244160204489984

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