Properties

Label 2-825-1.1-c5-0-117
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  + 5.42·2-s − 9·3-s − 2.52·4-s − 48.8·6-s − 29.0·7-s − 187.·8-s + 81·9-s + 121·11-s + 22.7·12-s − 433.·13-s − 157.·14-s − 936.·16-s + 681.·17-s + 439.·18-s + 1.40e3·19-s + 261.·21-s + 656.·22-s + 4.76e3·23-s + 1.68e3·24-s − 2.35e3·26-s − 729·27-s + 73.5·28-s − 646.·29-s + 840.·31-s + 913.·32-s − 1.08e3·33-s + 3.70e3·34-s + ⋯
L(s)  = 1  + 0.959·2-s − 0.577·3-s − 0.0789·4-s − 0.554·6-s − 0.224·7-s − 1.03·8-s + 0.333·9-s + 0.301·11-s + 0.0456·12-s − 0.710·13-s − 0.215·14-s − 0.914·16-s + 0.572·17-s + 0.319·18-s + 0.889·19-s + 0.129·21-s + 0.289·22-s + 1.87·23-s + 0.597·24-s − 0.682·26-s − 0.192·27-s + 0.0177·28-s − 0.142·29-s + 0.157·31-s + 0.157·32-s − 0.174·33-s + 0.548·34-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 \( 1 - 5.42T + 32T^{2} \)
7 \( 1 + 29.0T + 1.68e4T^{2} \)
13 \( 1 + 433.T + 3.71e5T^{2} \)
17 \( 1 - 681.T + 1.41e6T^{2} \)
19 \( 1 - 1.40e3T + 2.47e6T^{2} \)
23 \( 1 - 4.76e3T + 6.43e6T^{2} \)
29 \( 1 + 646.T + 2.05e7T^{2} \)
31 \( 1 - 840.T + 2.86e7T^{2} \)
37 \( 1 - 7.78e3T + 6.93e7T^{2} \)
41 \( 1 + 1.57e4T + 1.15e8T^{2} \)
43 \( 1 + 857.T + 1.47e8T^{2} \)
47 \( 1 + 9.40e3T + 2.29e8T^{2} \)
53 \( 1 + 3.88e4T + 4.18e8T^{2} \)
59 \( 1 - 4.33e4T + 7.14e8T^{2} \)
61 \( 1 + 3.56e4T + 8.44e8T^{2} \)
67 \( 1 + 4.28e3T + 1.35e9T^{2} \)
71 \( 1 - 790.T + 1.80e9T^{2} \)
73 \( 1 - 5.43e4T + 2.07e9T^{2} \)
79 \( 1 - 2.46e4T + 3.07e9T^{2} \)
83 \( 1 - 6.77e4T + 3.93e9T^{2} \)
89 \( 1 + 4.30e4T + 5.58e9T^{2} \)
97 \( 1 + 7.72e4T + 8.58e9T^{2} \)
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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

−9.335314043808248795995321291622, −8.111286130525201111987352163168, −7.01165370575087308855466579378, −6.28336766342584010270888236415, −5.19450018928501521912654058342, −4.86108713939122607354869376814, −3.62095086916265905725291424782, −2.84189265127606840548399828341, −1.18888129072476561221240475386, 0, 1.18888129072476561221240475386, 2.84189265127606840548399828341, 3.62095086916265905725291424782, 4.86108713939122607354869376814, 5.19450018928501521912654058342, 6.28336766342584010270888236415, 7.01165370575087308855466579378, 8.111286130525201111987352163168, 9.335314043808248795995321291622

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