Properties

Label 2-825-1.1-c1-0-30
Degree $2$
Conductor $825$
Sign $-1$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s − 3-s + 0.999·4-s − 1.73·6-s − 2·7-s − 1.73·8-s + 9-s − 11-s − 0.999·12-s − 5.46·13-s − 3.46·14-s − 5·16-s + 1.73·18-s + 5.46·19-s + 2·21-s − 1.73·22-s − 6.92·23-s + 1.73·24-s − 9.46·26-s − 27-s − 1.99·28-s − 3.46·29-s − 10.9·31-s − 5.19·32-s + 33-s + 0.999·36-s + 4.92·37-s + ⋯
L(s)  = 1  + 1.22·2-s − 0.577·3-s + 0.499·4-s − 0.707·6-s − 0.755·7-s − 0.612·8-s + 0.333·9-s − 0.301·11-s − 0.288·12-s − 1.51·13-s − 0.925·14-s − 1.25·16-s + 0.408·18-s + 1.25·19-s + 0.436·21-s − 0.369·22-s − 1.44·23-s + 0.353·24-s − 1.85·26-s − 0.192·27-s − 0.377·28-s − 0.643·29-s − 1.96·31-s − 0.918·32-s + 0.174·33-s + 0.166·36-s + 0.810·37-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: no
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 \( 1 - 1.73T + 2T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
13 \( 1 + 5.46T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 5.46T + 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 - 4.92T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 4.92T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 0.928T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 8.39T + 73T^{2} \)
79 \( 1 + 6.53T + 79T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 - 0.928T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−9.686714701389686651026132086190, −9.370474085062089692376184232137, −7.72974972916437071272652813370, −7.01502307853168229988256702430, −5.89127471605424311936608355157, −5.42177654495458161759189796888, −4.43066430794526292534749169160, −3.49179412737773871035965089826, −2.37648288046462023441843942037, 0, 2.37648288046462023441843942037, 3.49179412737773871035965089826, 4.43066430794526292534749169160, 5.42177654495458161759189796888, 5.89127471605424311936608355157, 7.01502307853168229988256702430, 7.72974972916437071272652813370, 9.370474085062089692376184232137, 9.686714701389686651026132086190

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