Properties

Label 2-825-1.1-c1-0-7
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.21·2-s − 3-s − 0.525·4-s + 1.21·6-s + 4.90·7-s + 3.06·8-s + 9-s − 11-s + 0.525·12-s + 4.14·13-s − 5.95·14-s − 2.67·16-s − 5.33·17-s − 1.21·18-s − 5.18·19-s − 4.90·21-s + 1.21·22-s + 4·23-s − 3.06·24-s − 5.03·26-s − 27-s − 2.57·28-s + 1.80·29-s + 2.62·31-s − 2.88·32-s + 33-s + 6.47·34-s + ⋯
L(s)  = 1  − 0.858·2-s − 0.577·3-s − 0.262·4-s + 0.495·6-s + 1.85·7-s + 1.08·8-s + 0.333·9-s − 0.301·11-s + 0.151·12-s + 1.15·13-s − 1.59·14-s − 0.668·16-s − 1.29·17-s − 0.286·18-s − 1.18·19-s − 1.06·21-s + 0.258·22-s + 0.834·23-s − 0.625·24-s − 0.987·26-s − 0.192·27-s − 0.486·28-s + 0.335·29-s + 0.470·31-s − 0.510·32-s + 0.174·33-s + 1.11·34-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: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8905407563\)
\(L(\frac12)\) \(\approx\) \(0.8905407563\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.21T + 2T^{2} \)
7 \( 1 - 4.90T + 7T^{2} \)
13 \( 1 - 4.14T + 13T^{2} \)
17 \( 1 + 5.33T + 17T^{2} \)
19 \( 1 + 5.18T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 1.80T + 29T^{2} \)
31 \( 1 - 2.62T + 31T^{2} \)
37 \( 1 - 5.80T + 37T^{2} \)
41 \( 1 - 1.80T + 41T^{2} \)
43 \( 1 - 4.90T + 43T^{2} \)
47 \( 1 - 7.05T + 47T^{2} \)
53 \( 1 + 7.18T + 53T^{2} \)
59 \( 1 - 1.67T + 59T^{2} \)
61 \( 1 - 0.755T + 61T^{2} \)
67 \( 1 + 4.85T + 67T^{2} \)
71 \( 1 - 0.428T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + 6.42T + 79T^{2} \)
83 \( 1 + 2.90T + 83T^{2} \)
89 \( 1 - 0.622T + 89T^{2} \)
97 \( 1 + 2.75T + 97T^{2} \)
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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

−10.45481668105131590752179542422, −9.168836514074333158517830630843, −8.511226800400000433009649511041, −7.978377094133140898193965544623, −6.96569658306558056074072397097, −5.81889204724798485412087037558, −4.67666615259883987453194053627, −4.28930424590553349006082575792, −2.08486401237106693669829180021, −0.963447407139270182895892958408, 0.963447407139270182895892958408, 2.08486401237106693669829180021, 4.28930424590553349006082575792, 4.67666615259883987453194053627, 5.81889204724798485412087037558, 6.96569658306558056074072397097, 7.978377094133140898193965544623, 8.511226800400000433009649511041, 9.168836514074333158517830630843, 10.45481668105131590752179542422

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