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  + 1.15·2-s − 0.220·3-s − 0.659·4-s − 0.421·5-s − 0.255·6-s − 0.645·7-s − 3.07·8-s − 2.95·9-s − 0.487·10-s − 0.640·11-s + 0.145·12-s − 4.07·13-s − 0.747·14-s + 0.0929·15-s − 2.24·16-s + 6.47·17-s − 3.41·18-s − 4.15·19-s + 0.277·20-s + 0.142·21-s − 0.741·22-s − 7.48·23-s + 0.679·24-s − 4.82·25-s − 4.71·26-s + 1.31·27-s + 0.425·28-s + ⋯
L(s)  = 1  + 0.818·2-s − 0.127·3-s − 0.329·4-s − 0.188·5-s − 0.104·6-s − 0.243·7-s − 1.08·8-s − 0.983·9-s − 0.154·10-s − 0.192·11-s + 0.0419·12-s − 1.12·13-s − 0.199·14-s + 0.0240·15-s − 0.561·16-s + 1.57·17-s − 0.805·18-s − 0.952·19-s + 0.0621·20-s + 0.0310·21-s − 0.158·22-s − 1.56·23-s + 0.138·24-s − 0.964·25-s − 0.924·26-s + 0.252·27-s + 0.0803·28-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 \( 1 - 1.15T + 2T^{2} \)
3 \( 1 + 0.220T + 3T^{2} \)
5 \( 1 + 0.421T + 5T^{2} \)
7 \( 1 + 0.645T + 7T^{2} \)
11 \( 1 + 0.640T + 11T^{2} \)
13 \( 1 + 4.07T + 13T^{2} \)
17 \( 1 - 6.47T + 17T^{2} \)
19 \( 1 + 4.15T + 19T^{2} \)
23 \( 1 + 7.48T + 23T^{2} \)
29 \( 1 + 0.298T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 6.30T + 37T^{2} \)
41 \( 1 - 3.79T + 41T^{2} \)
43 \( 1 - 0.987T + 43T^{2} \)
47 \( 1 - 0.803T + 47T^{2} \)
53 \( 1 + 0.867T + 53T^{2} \)
59 \( 1 - 0.841T + 59T^{2} \)
61 \( 1 + 5.88T + 61T^{2} \)
67 \( 1 - 3.61T + 67T^{2} \)
71 \( 1 + 4.80T + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 - 5.23T + 79T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 + 9.06T + 89T^{2} \)
97 \( 1 - 5.13T + 97T^{2} \)
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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

−10.23068804261339461236653121268, −9.651639831990040984906736503952, −8.455916343871390903655098621179, −7.77925729697143064995723929153, −6.30983924619455735658012761867, −5.63556699067308146067789074196, −4.69144464869815195290085995252, −3.62796897450870693902336149362, −2.57288271879330785701893290617, 0, 2.57288271879330785701893290617, 3.62796897450870693902336149362, 4.69144464869815195290085995252, 5.63556699067308146067789074196, 6.30983924619455735658012761867, 7.77925729697143064995723929153, 8.455916343871390903655098621179, 9.651639831990040984906736503952, 10.23068804261339461236653121268

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