Properties

Label 2-825-1.1-c3-0-5
Degree $2$
Conductor $825$
Sign $1$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s − 3·3-s + 17·4-s + 15·6-s − 3·7-s − 45·8-s + 9·9-s − 11·11-s − 51·12-s − 32·13-s + 15·14-s + 89·16-s − 33·17-s − 45·18-s + 47·19-s + 9·21-s + 55·22-s − 113·23-s + 135·24-s + 160·26-s − 27·27-s − 51·28-s − 54·29-s + 178·31-s − 85·32-s + 33·33-s + 165·34-s + ⋯
L(s)  = 1  − 1.76·2-s − 0.577·3-s + 17/8·4-s + 1.02·6-s − 0.161·7-s − 1.98·8-s + 1/3·9-s − 0.301·11-s − 1.22·12-s − 0.682·13-s + 0.286·14-s + 1.39·16-s − 0.470·17-s − 0.589·18-s + 0.567·19-s + 0.0935·21-s + 0.533·22-s − 1.02·23-s + 1.14·24-s + 1.20·26-s − 0.192·27-s − 0.344·28-s − 0.345·29-s + 1.03·31-s − 0.469·32-s + 0.174·33-s + 0.832·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3848540493\)
\(L(\frac12)\) \(\approx\) \(0.3848540493\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p T \)
5 \( 1 \)
11 \( 1 + p T \)
good2 \( 1 + 5 T + p^{3} T^{2} \)
7 \( 1 + 3 T + p^{3} T^{2} \)
13 \( 1 + 32 T + p^{3} T^{2} \)
17 \( 1 + 33 T + p^{3} T^{2} \)
19 \( 1 - 47 T + p^{3} T^{2} \)
23 \( 1 + 113 T + p^{3} T^{2} \)
29 \( 1 + 54 T + p^{3} T^{2} \)
31 \( 1 - 178 T + p^{3} T^{2} \)
37 \( 1 + 19 T + p^{3} T^{2} \)
41 \( 1 - 139 T + p^{3} T^{2} \)
43 \( 1 - 308 T + p^{3} T^{2} \)
47 \( 1 + 195 T + p^{3} T^{2} \)
53 \( 1 + 152 T + p^{3} T^{2} \)
59 \( 1 + 625 T + p^{3} T^{2} \)
61 \( 1 - 320 T + p^{3} T^{2} \)
67 \( 1 + 200 T + p^{3} T^{2} \)
71 \( 1 + 947 T + p^{3} T^{2} \)
73 \( 1 - 448 T + p^{3} T^{2} \)
79 \( 1 + 721 T + p^{3} T^{2} \)
83 \( 1 + 142 T + p^{3} T^{2} \)
89 \( 1 - 404 T + p^{3} T^{2} \)
97 \( 1 + 79 T + p^{3} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.811021672104947389089780926873, −9.164068148390571035206558399274, −8.115806408807401130792124009830, −7.50506055730627607780729180908, −6.64322727489192114310104377908, −5.80548181248261029059555322521, −4.51068218999076718561587436022, −2.84679337150766418326259469437, −1.72245949754762610921239561276, −0.45779001117160984669781443764, 0.45779001117160984669781443764, 1.72245949754762610921239561276, 2.84679337150766418326259469437, 4.51068218999076718561587436022, 5.80548181248261029059555322521, 6.64322727489192114310104377908, 7.50506055730627607780729180908, 8.115806408807401130792124009830, 9.164068148390571035206558399274, 9.811021672104947389089780926873

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