Properties

Label 1-547-547.464-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.656 - 0.754i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.845 − 0.534i)2-s + 3-s + (0.428 + 0.903i)4-s + (−0.919 − 0.391i)5-s + (−0.845 − 0.534i)6-s + (−0.0402 − 0.999i)7-s + (0.120 − 0.992i)8-s + 9-s + (0.568 + 0.822i)10-s + (−0.200 + 0.979i)11-s + (0.428 + 0.903i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.919 − 0.391i)15-s + (−0.632 + 0.774i)16-s + (0.948 + 0.316i)17-s + ⋯
L(s)  = 1  + (−0.845 − 0.534i)2-s + 3-s + (0.428 + 0.903i)4-s + (−0.919 − 0.391i)5-s + (−0.845 − 0.534i)6-s + (−0.0402 − 0.999i)7-s + (0.120 − 0.992i)8-s + 9-s + (0.568 + 0.822i)10-s + (−0.200 + 0.979i)11-s + (0.428 + 0.903i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.919 − 0.391i)15-s + (−0.632 + 0.774i)16-s + (0.948 + 0.316i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.656 - 0.754i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (464, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.656 - 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3623237112 - 0.7960650402i\)
\(L(\frac12)\) \(\approx\) \(0.3623237112 - 0.7960650402i\)
\(L(1)\) \(\approx\) \(0.7142516036 - 0.3895504666i\)
\(L(1)\) \(\approx\) \(0.7142516036 - 0.3895504666i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.845 - 0.534i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.919 - 0.391i)T \)
7 \( 1 + (-0.0402 - 0.999i)T \)
11 \( 1 + (-0.200 + 0.979i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.948 + 0.316i)T \)
19 \( 1 + (-0.632 - 0.774i)T \)
23 \( 1 + (0.278 - 0.960i)T \)
29 \( 1 + (-0.970 - 0.239i)T \)
31 \( 1 + (-0.354 - 0.935i)T \)
37 \( 1 + (0.278 + 0.960i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.996 - 0.0804i)T \)
47 \( 1 + (0.692 - 0.721i)T \)
53 \( 1 + (0.799 - 0.600i)T \)
59 \( 1 + (-0.632 + 0.774i)T \)
61 \( 1 + (-0.845 - 0.534i)T \)
67 \( 1 + (-0.996 - 0.0804i)T \)
71 \( 1 + (0.987 - 0.160i)T \)
73 \( 1 + (-0.919 + 0.391i)T \)
79 \( 1 + (0.568 - 0.822i)T \)
83 \( 1 + (0.799 - 0.600i)T \)
89 \( 1 + (0.885 + 0.464i)T \)
97 \( 1 + (-0.200 - 0.979i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.89451562245769489148050504953, −23.22429685435343155121812776764, −21.78118660539800686509981091226, −21.12716869784962738361782306438, −19.962405168576078295157492182529, −19.20060443333608182024069427766, −18.81244198998475479483912198613, −18.22565022716371311321270191794, −16.61017665935898124136102684241, −16.12155986081141988301906868290, −15.14596264531877852262524300112, −14.68016601462301209973992447396, −13.86285100705172500058277986472, −12.4129101115818581341605368334, −11.53588648507186686920500214805, −10.552411172869170798364187901727, −9.43253242582959578580752770766, −8.81127463782647179249928411872, −7.95390061602916722812464285028, −7.34485679115517090659395944146, −6.24175743353969368137007940446, −5.09110729250885798390876727314, −3.62588043073611678984014838416, −2.69699189997790844108536770395, −1.53497539659846489435231287000, 0.541618113117078756862659129374, 1.85329000075227183678079042123, 3.02649934958370124363411878978, 3.898054545253233836652724776847, 4.7051870855486186872601529998, 6.93567853164927828342276909321, 7.57193698859859891535983105008, 8.13101112433958000081406495794, 9.11097891138666789125805107511, 10.0968600532941492923636690749, 10.63706354370980744394019328589, 11.95200036186477182931431145371, 12.77882379263409315086547216040, 13.34234176775082730726898500436, 14.961495701726439951906584251177, 15.2632985088798479196374328953, 16.58305449251661136878094130802, 17.081253001263120346388332806054, 18.300023574304151097656583024525, 19.10618751054587991313757913625, 19.84014783635343784964566547676, 20.43082710951951431357182157706, 20.760419641434751868648558435, 22.080084472103364809359420069928, 23.11315703873232156243556999556

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