Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.523·2-s − 3.08·3-s − 1.72·4-s − 1.35·5-s + 1.61·6-s + 3.60·7-s + 1.95·8-s + 6.53·9-s + 0.707·10-s − 0.337·11-s + 5.32·12-s − 0.968·13-s − 1.88·14-s + 4.17·15-s + 2.43·16-s + 1.24·17-s − 3.41·18-s − 5.50·19-s + 2.33·20-s − 11.1·21-s + 0.176·22-s + 3.13·23-s − 6.02·24-s − 3.17·25-s + 0.506·26-s − 10.9·27-s − 6.22·28-s + ⋯
L(s)  = 1  − 0.370·2-s − 1.78·3-s − 0.862·4-s − 0.604·5-s + 0.659·6-s + 1.36·7-s + 0.689·8-s + 2.17·9-s + 0.223·10-s − 0.101·11-s + 1.53·12-s − 0.268·13-s − 0.504·14-s + 1.07·15-s + 0.607·16-s + 0.302·17-s − 0.805·18-s − 1.26·19-s + 0.521·20-s − 2.43·21-s + 0.0377·22-s + 0.653·23-s − 1.22·24-s − 0.634·25-s + 0.0993·26-s − 2.09·27-s − 1.17·28-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 \( 1 + 0.523T + 2T^{2} \)
3 \( 1 + 3.08T + 3T^{2} \)
5 \( 1 + 1.35T + 5T^{2} \)
7 \( 1 - 3.60T + 7T^{2} \)
11 \( 1 + 0.337T + 11T^{2} \)
13 \( 1 + 0.968T + 13T^{2} \)
17 \( 1 - 1.24T + 17T^{2} \)
19 \( 1 + 5.50T + 19T^{2} \)
23 \( 1 - 3.13T + 23T^{2} \)
29 \( 1 + 7.18T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 5.45T + 37T^{2} \)
41 \( 1 + 7.24T + 41T^{2} \)
43 \( 1 + 6.48T + 43T^{2} \)
47 \( 1 + 7.68T + 47T^{2} \)
53 \( 1 + 4.50T + 53T^{2} \)
59 \( 1 - 2.41T + 59T^{2} \)
61 \( 1 - 6.37T + 61T^{2} \)
67 \( 1 - 0.799T + 67T^{2} \)
71 \( 1 - 2.93T + 71T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 4.42T + 89T^{2} \)
97 \( 1 + 6.98T + 97T^{2} \)
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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

−10.49268244347327777404024106786, −9.751647872167011370004899487361, −8.385975871355432415294033212927, −7.78814116227222838822740224339, −6.67540067255841058577513289577, −5.46842249999263062779131220481, −4.77974251600609350804068680944, −4.12972175387888128951856354125, −1.43648284305103587760472889877, 0, 1.43648284305103587760472889877, 4.12972175387888128951856354125, 4.77974251600609350804068680944, 5.46842249999263062779131220481, 6.67540067255841058577513289577, 7.78814116227222838822740224339, 8.385975871355432415294033212927, 9.751647872167011370004899487361, 10.49268244347327777404024106786

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