Properties

Degree $2$
Conductor $547$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.35·2-s − 3.09·3-s + 3.56·4-s − 4.18·5-s + 7.29·6-s − 2.82·7-s − 3.69·8-s + 6.54·9-s + 9.86·10-s + 2.48·11-s − 11.0·12-s + 4.60·13-s + 6.67·14-s + 12.9·15-s + 1.59·16-s − 5.70·17-s − 15.4·18-s + 0.542·19-s − 14.9·20-s + 8.74·21-s − 5.86·22-s + 6.30·23-s + 11.4·24-s + 12.4·25-s − 10.8·26-s − 10.9·27-s − 10.0·28-s + ⋯
L(s)  = 1  − 1.66·2-s − 1.78·3-s + 1.78·4-s − 1.87·5-s + 2.97·6-s − 1.06·7-s − 1.30·8-s + 2.18·9-s + 3.12·10-s + 0.749·11-s − 3.18·12-s + 1.27·13-s + 1.78·14-s + 3.33·15-s + 0.397·16-s − 1.38·17-s − 3.64·18-s + 0.124·19-s − 3.33·20-s + 1.90·21-s − 1.25·22-s + 1.31·23-s + 2.33·24-s + 2.49·25-s − 2.12·26-s − 2.11·27-s − 1.90·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(547\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{547} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 547,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 + T \)
good2 \( 1 + 2.35T + 2T^{2} \)
3 \( 1 + 3.09T + 3T^{2} \)
5 \( 1 + 4.18T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 - 2.48T + 11T^{2} \)
13 \( 1 - 4.60T + 13T^{2} \)
17 \( 1 + 5.70T + 17T^{2} \)
19 \( 1 - 0.542T + 19T^{2} \)
23 \( 1 - 6.30T + 23T^{2} \)
29 \( 1 + 4.16T + 29T^{2} \)
31 \( 1 - 5.03T + 31T^{2} \)
37 \( 1 + 4.02T + 37T^{2} \)
41 \( 1 + 2.04T + 41T^{2} \)
43 \( 1 - 7.11T + 43T^{2} \)
47 \( 1 + 4.23T + 47T^{2} \)
53 \( 1 + 8.60T + 53T^{2} \)
59 \( 1 - 2.15T + 59T^{2} \)
61 \( 1 + 8.53T + 61T^{2} \)
67 \( 1 + 3.63T + 67T^{2} \)
71 \( 1 - 7.43T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 - 4.82T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 - 2.44T + 89T^{2} \)
97 \( 1 - 5.35T + 97T^{2} \)
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

−10.76166949042289388187837409603, −9.434270636438326087606800012568, −8.706779270795272429277612404325, −7.60852842286829346138178098350, −6.66860478662728389063772173467, −6.45872582097565305774566668301, −4.66547788379470256907594808681, −3.56994162980590643847882165835, −1.00546465244164347558091991985, 0, 1.00546465244164347558091991985, 3.56994162980590643847882165835, 4.66547788379470256907594808681, 6.45872582097565305774566668301, 6.66860478662728389063772173467, 7.60852842286829346138178098350, 8.706779270795272429277612404325, 9.434270636438326087606800012568, 10.76166949042289388187837409603

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