Properties

Label 2-6525-1.1-c1-0-141
Degree $2$
Conductor $6525$
Sign $-1$
Analytic cond. $52.1023$
Root an. cond. $7.21819$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.75·2-s + 5.59·4-s − 0.393·7-s − 9.92·8-s + 0.393·11-s + 2.56·13-s + 1.08·14-s + 16.1·16-s + 2.07·17-s − 0.958·19-s − 1.08·22-s + 6.15·23-s − 7.07·26-s − 2.20·28-s + 29-s − 10.1·31-s − 24.6·32-s − 5.72·34-s + 7.34·37-s + 2.64·38-s + 1.65·41-s − 10.3·43-s + 2.20·44-s − 16.9·46-s − 11.5·47-s − 6.84·49-s + 14.3·52-s + ⋯
L(s)  = 1  − 1.94·2-s + 2.79·4-s − 0.148·7-s − 3.50·8-s + 0.118·11-s + 0.711·13-s + 0.290·14-s + 4.03·16-s + 0.504·17-s − 0.219·19-s − 0.231·22-s + 1.28·23-s − 1.38·26-s − 0.416·28-s + 0.185·29-s − 1.82·31-s − 4.36·32-s − 0.982·34-s + 1.20·37-s + 0.428·38-s + 0.258·41-s − 1.57·43-s + 0.332·44-s − 2.50·46-s − 1.69·47-s − 0.977·49-s + 1.99·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6525\)    =    \(3^{2} \cdot 5^{2} \cdot 29\)
Sign: $-1$
Analytic conductor: \(52.1023\)
Root analytic conductor: \(7.21819\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6525,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
29 \( 1 - T \)
good2 \( 1 + 2.75T + 2T^{2} \)
7 \( 1 + 0.393T + 7T^{2} \)
11 \( 1 - 0.393T + 11T^{2} \)
13 \( 1 - 2.56T + 13T^{2} \)
17 \( 1 - 2.07T + 17T^{2} \)
19 \( 1 + 0.958T + 19T^{2} \)
23 \( 1 - 6.15T + 23T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 - 7.34T + 37T^{2} \)
41 \( 1 - 1.65T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 - 9.54T + 59T^{2} \)
61 \( 1 + 6.25T + 61T^{2} \)
67 \( 1 - 7.42T + 67T^{2} \)
71 \( 1 + 5.98T + 71T^{2} \)
73 \( 1 + 3.34T + 73T^{2} \)
79 \( 1 + 2.06T + 79T^{2} \)
83 \( 1 - 6.41T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 - 18.4T + 97T^{2} \)
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

−7.903944537365941604712567684300, −7.12536036562838864979297773676, −6.55448738133037053387209997695, −5.94314820029280905918238623542, −4.99338835440318987267548504548, −3.53617766749138504681237536357, −2.97386712902853447559633114746, −1.84816694429451842278304496902, −1.17288904846552800385341163843, 0, 1.17288904846552800385341163843, 1.84816694429451842278304496902, 2.97386712902853447559633114746, 3.53617766749138504681237536357, 4.99338835440318987267548504548, 5.94314820029280905918238623542, 6.55448738133037053387209997695, 7.12536036562838864979297773676, 7.903944537365941604712567684300

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