Properties

Label 1-547-547.402-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: $\chi_{547} (402, \cdot )$
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

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

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