Properties

Label 2-3549-1.1-c1-0-74
Degree $2$
Conductor $3549$
Sign $1$
Analytic cond. $28.3389$
Root an. cond. $5.32343$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.67·2-s − 3-s + 5.14·4-s − 1.32·5-s − 2.67·6-s − 7-s + 8.39·8-s + 9-s − 3.54·10-s − 3.03·11-s − 5.14·12-s − 2.67·14-s + 1.32·15-s + 12.1·16-s + 4.56·17-s + 2.67·18-s + 0.940·19-s − 6.81·20-s + 21-s − 8.10·22-s + 8.55·23-s − 8.39·24-s − 3.24·25-s − 27-s − 5.14·28-s + 5.79·29-s + 3.54·30-s + ⋯
L(s)  = 1  + 1.88·2-s − 0.577·3-s + 2.57·4-s − 0.592·5-s − 1.09·6-s − 0.377·7-s + 2.96·8-s + 0.333·9-s − 1.12·10-s − 0.914·11-s − 1.48·12-s − 0.714·14-s + 0.342·15-s + 3.03·16-s + 1.10·17-s + 0.629·18-s + 0.215·19-s − 1.52·20-s + 0.218·21-s − 1.72·22-s + 1.78·23-s − 1.71·24-s − 0.648·25-s − 0.192·27-s − 0.971·28-s + 1.07·29-s + 0.646·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(28.3389\)
Root analytic conductor: \(5.32343\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.875657291\)
\(L(\frac12)\) \(\approx\) \(4.875657291\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - 2.67T + 2T^{2} \)
5 \( 1 + 1.32T + 5T^{2} \)
11 \( 1 + 3.03T + 11T^{2} \)
17 \( 1 - 4.56T + 17T^{2} \)
19 \( 1 - 0.940T + 19T^{2} \)
23 \( 1 - 8.55T + 23T^{2} \)
29 \( 1 - 5.79T + 29T^{2} \)
31 \( 1 - 1.00T + 31T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 + 7.71T + 41T^{2} \)
43 \( 1 - 7.20T + 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 - 5.26T + 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 - 5.82T + 61T^{2} \)
67 \( 1 - 3.02T + 67T^{2} \)
71 \( 1 + 6.04T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 - 1.46T + 79T^{2} \)
83 \( 1 - 4.31T + 83T^{2} \)
89 \( 1 + 3.22T + 89T^{2} \)
97 \( 1 + 5.51T + 97T^{2} \)
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

−8.145744633347228274537530819924, −7.42386572094822953127338468038, −6.93385640264947015580935375116, −5.95461805272815655472022756376, −5.52477685716629592203364250299, −4.73424368153139595132488182418, −4.10198665114468585642593501732, −3.14473230843182922620395825715, −2.60630318261803059938219132984, −1.04717610594165875993501620301, 1.04717610594165875993501620301, 2.60630318261803059938219132984, 3.14473230843182922620395825715, 4.10198665114468585642593501732, 4.73424368153139595132488182418, 5.52477685716629592203364250299, 5.95461805272815655472022756376, 6.93385640264947015580935375116, 7.42386572094822953127338468038, 8.145744633347228274537530819924

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