Properties

Label 2-825-1.1-c1-0-18
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  + 2.70·2-s − 3-s + 5.34·4-s − 2.70·6-s − 1.07·7-s + 9.04·8-s + 9-s + 11-s − 5.34·12-s + 4.34·13-s − 2.92·14-s + 13.8·16-s − 7.75·17-s + 2.70·18-s + 5.26·19-s + 1.07·21-s + 2.70·22-s + 2.15·23-s − 9.04·24-s + 11.7·26-s − 27-s − 5.75·28-s + 1.41·29-s − 4.68·31-s + 19.3·32-s − 33-s − 21.0·34-s + ⋯
L(s)  = 1  + 1.91·2-s − 0.577·3-s + 2.67·4-s − 1.10·6-s − 0.407·7-s + 3.19·8-s + 0.333·9-s + 0.301·11-s − 1.54·12-s + 1.20·13-s − 0.780·14-s + 3.45·16-s − 1.88·17-s + 0.638·18-s + 1.20·19-s + 0.235·21-s + 0.577·22-s + 0.449·23-s − 1.84·24-s + 2.30·26-s − 0.192·27-s − 1.08·28-s + 0.263·29-s − 0.840·31-s + 3.42·32-s − 0.174·33-s − 3.60·34-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

\(L(1)\) \(\approx\) \(4.256474993\)
\(L(\frac12)\) \(\approx\) \(4.256474993\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.70T + 2T^{2} \)
7 \( 1 + 1.07T + 7T^{2} \)
13 \( 1 - 4.34T + 13T^{2} \)
17 \( 1 + 7.75T + 17T^{2} \)
19 \( 1 - 5.26T + 19T^{2} \)
23 \( 1 - 2.15T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 4.68T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 9.41T + 41T^{2} \)
43 \( 1 + 7.60T + 43T^{2} \)
47 \( 1 + 4.68T + 47T^{2} \)
53 \( 1 + 0.156T + 53T^{2} \)
59 \( 1 - 6.15T + 59T^{2} \)
61 \( 1 + 4.15T + 61T^{2} \)
67 \( 1 - 8.68T + 67T^{2} \)
71 \( 1 + 4.68T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + 8.09T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−10.82709174380023395352038537089, −9.630798721978810368269117603607, −8.318272012970896134030915355471, −6.86355015065206450571995951020, −6.65996652849161641384712338128, −5.66318824732485087487866037554, −4.88109582633404357410849875738, −3.93947098832744351545324993545, −3.10743837981281914408068596125, −1.67763356661225371788350724459, 1.67763356661225371788350724459, 3.10743837981281914408068596125, 3.93947098832744351545324993545, 4.88109582633404357410849875738, 5.66318824732485087487866037554, 6.65996652849161641384712338128, 6.86355015065206450571995951020, 8.318272012970896134030915355471, 9.630798721978810368269117603607, 10.82709174380023395352038537089

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