Properties

Label 2-825-1.1-c1-0-19
Degree $2$
Conductor $825$
Sign $-1$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s + 7-s + 9-s − 11-s + 2·12-s + 13-s + 4·16-s + 6·17-s − 7·19-s − 21-s − 6·23-s − 27-s − 2·28-s − 6·29-s − 7·31-s + 33-s − 2·36-s − 2·37-s − 39-s − 6·41-s + 43-s + 2·44-s − 4·48-s − 6·49-s − 6·51-s − 2·52-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.577·12-s + 0.277·13-s + 16-s + 1.45·17-s − 1.60·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s − 0.377·28-s − 1.11·29-s − 1.25·31-s + 0.174·33-s − 1/3·36-s − 0.328·37-s − 0.160·39-s − 0.937·41-s + 0.152·43-s + 0.301·44-s − 0.577·48-s − 6/7·49-s − 0.840·51-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{825} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
11 \( 1 + T \)
good2 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−9.974328952037568305037681431369, −8.914517223015984535585916849605, −8.166540824143412625936193267718, −7.35087717315634547606388680691, −5.99232217815539757223706684675, −5.41932401993915498989382893015, −4.39119879142282475890989547542, −3.55718547289049389773277815438, −1.70032221690921013211384714862, 0, 1.70032221690921013211384714862, 3.55718547289049389773277815438, 4.39119879142282475890989547542, 5.41932401993915498989382893015, 5.99232217815539757223706684675, 7.35087717315634547606388680691, 8.166540824143412625936193267718, 8.914517223015984535585916849605, 9.974328952037568305037681431369

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