Properties

Label 2-91e2-1.1-c1-0-141
Degree $2$
Conductor $8281$
Sign $1$
Analytic cond. $66.1241$
Root an. cond. $8.13167$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 3·5-s − 3·9-s − 6·10-s + 6·11-s − 4·16-s − 4·17-s − 6·18-s + 5·19-s − 6·20-s + 12·22-s + 3·23-s + 4·25-s − 5·29-s − 3·31-s − 8·32-s − 8·34-s − 6·36-s + 4·37-s + 10·38-s − 6·41-s − 43-s + 12·44-s + 9·45-s + 6·46-s + 7·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.34·5-s − 9-s − 1.89·10-s + 1.80·11-s − 16-s − 0.970·17-s − 1.41·18-s + 1.14·19-s − 1.34·20-s + 2.55·22-s + 0.625·23-s + 4/5·25-s − 0.928·29-s − 0.538·31-s − 1.41·32-s − 1.37·34-s − 36-s + 0.657·37-s + 1.62·38-s − 0.937·41-s − 0.152·43-s + 1.80·44-s + 1.34·45-s + 0.884·46-s + 1.02·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8281\)    =    \(7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(66.1241\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8281} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8281,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.799788347\)
\(L(\frac12)\) \(\approx\) \(2.799788347\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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

−7.57740679946665025654456265521, −6.90678100068040939590507024634, −6.38026602914856538020136886750, −5.57634080836583272363145209435, −4.89731593269989771358095794547, −4.14522542139967607027503862649, −3.61956902475371613886505030678, −3.18335929774170366064894969897, −2.07697976816031338635613740258, −0.64616466383857911844066535116, 0.64616466383857911844066535116, 2.07697976816031338635613740258, 3.18335929774170366064894969897, 3.61956902475371613886505030678, 4.14522542139967607027503862649, 4.89731593269989771358095794547, 5.57634080836583272363145209435, 6.38026602914856538020136886750, 6.90678100068040939590507024634, 7.57740679946665025654456265521

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