Properties

Label 2-825-1.1-c1-0-3
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  − 2.12·2-s + 3-s + 2.51·4-s − 2.12·6-s − 3.64·7-s − 1.09·8-s + 9-s + 11-s + 2.51·12-s − 1.51·13-s + 7.73·14-s − 2.70·16-s − 1.15·17-s − 2.12·18-s + 2.60·19-s − 3.64·21-s − 2.12·22-s + 5.73·23-s − 1.09·24-s + 3.21·26-s + 27-s − 9.15·28-s + 6.24·29-s + 5.51·31-s + 7.93·32-s + 33-s + 2.45·34-s + ⋯
L(s)  = 1  − 1.50·2-s + 0.577·3-s + 1.25·4-s − 0.867·6-s − 1.37·7-s − 0.387·8-s + 0.333·9-s + 0.301·11-s + 0.726·12-s − 0.420·13-s + 2.06·14-s − 0.676·16-s − 0.280·17-s − 0.500·18-s + 0.598·19-s − 0.794·21-s − 0.453·22-s + 1.19·23-s − 0.223·24-s + 0.631·26-s + 0.192·27-s − 1.73·28-s + 1.16·29-s + 0.990·31-s + 1.40·32-s + 0.174·33-s + 0.420·34-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

\(L(1)\) \(\approx\) \(0.7432211161\)
\(L(\frac12)\) \(\approx\) \(0.7432211161\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.12T + 2T^{2} \)
7 \( 1 + 3.64T + 7T^{2} \)
13 \( 1 + 1.51T + 13T^{2} \)
17 \( 1 + 1.15T + 17T^{2} \)
19 \( 1 - 2.60T + 19T^{2} \)
23 \( 1 - 5.73T + 23T^{2} \)
29 \( 1 - 6.24T + 29T^{2} \)
31 \( 1 - 5.51T + 31T^{2} \)
37 \( 1 + 0.454T + 37T^{2} \)
41 \( 1 - 4.12T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 - 3.48T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + 7.73T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 - 8.51T + 71T^{2} \)
73 \( 1 - 9.21T + 73T^{2} \)
79 \( 1 - 5.09T + 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 - 6.77T + 97T^{2} \)
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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

−9.943478405731546848986993393111, −9.354422200049265288522957312997, −8.758315747985887597800622329874, −7.86609369650864828657033254658, −6.95060357752092697923653229552, −6.43591438537674469624360095397, −4.81317744658448132552956067139, −3.39406231718069596091459456266, −2.43667557594614938349342832539, −0.853024506382851953622121402073, 0.853024506382851953622121402073, 2.43667557594614938349342832539, 3.39406231718069596091459456266, 4.81317744658448132552956067139, 6.43591438537674469624360095397, 6.95060357752092697923653229552, 7.86609369650864828657033254658, 8.758315747985887597800622329874, 9.354422200049265288522957312997, 9.943478405731546848986993393111

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