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Properties

Label 1-547-547.19-r0-0-0
Degree $1$
Conductor $547$
Sign $0.956 - 0.290i$
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  + (−0.880 + 0.474i)2-s + (−0.222 − 0.974i)3-s + (0.548 − 0.835i)4-s + (−0.819 + 0.572i)5-s + (0.658 + 0.752i)6-s + (−0.763 + 0.645i)7-s + (−0.0862 + 0.996i)8-s + (−0.900 + 0.433i)9-s + (0.449 − 0.893i)10-s + (−0.996 + 0.0804i)11-s + (−0.937 − 0.349i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (0.740 + 0.671i)15-s + (−0.397 − 0.917i)16-s + (−0.778 − 0.627i)17-s + ⋯
L(s)  = 1  + (−0.880 + 0.474i)2-s + (−0.222 − 0.974i)3-s + (0.548 − 0.835i)4-s + (−0.819 + 0.572i)5-s + (0.658 + 0.752i)6-s + (−0.763 + 0.645i)7-s + (−0.0862 + 0.996i)8-s + (−0.900 + 0.433i)9-s + (0.449 − 0.893i)10-s + (−0.996 + 0.0804i)11-s + (−0.937 − 0.349i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (0.740 + 0.671i)15-s + (−0.397 − 0.917i)16-s + (−0.778 − 0.627i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.956 - 0.290i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.956 - 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3372707509 - 0.05007374500i\)
\(L(\frac12)\) \(\approx\) \(0.3372707509 - 0.05007374500i\)
\(L(1)\) \(\approx\) \(0.4268513019 + 0.003786822019i\)
\(L(1)\) \(\approx\) \(0.4268513019 + 0.003786822019i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.880 + 0.474i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.819 + 0.572i)T \)
7 \( 1 + (-0.763 + 0.645i)T \)
11 \( 1 + (-0.996 + 0.0804i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
17 \( 1 + (-0.778 - 0.627i)T \)
19 \( 1 + (0.756 - 0.654i)T \)
23 \( 1 + (-0.890 + 0.454i)T \)
29 \( 1 + (0.962 - 0.272i)T \)
31 \( 1 + (-0.650 - 0.759i)T \)
37 \( 1 + (0.605 + 0.795i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.874 + 0.484i)T \)
47 \( 1 + (0.948 + 0.316i)T \)
53 \( 1 + (0.211 - 0.977i)T \)
59 \( 1 + (-0.0402 + 0.999i)T \)
61 \( 1 + (0.998 - 0.0460i)T \)
67 \( 1 + (0.166 + 0.986i)T \)
71 \( 1 + (0.529 - 0.848i)T \)
73 \( 1 + (0.740 - 0.671i)T \)
79 \( 1 + (-0.0172 - 0.999i)T \)
83 \( 1 + (-0.632 - 0.774i)T \)
89 \( 1 + (-0.539 + 0.842i)T \)
97 \( 1 + (-0.558 - 0.829i)T \)
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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

−23.34015742181861576491058262193, −22.366217727921959705480945203612, −21.66409615273548659701673732258, −20.61575803394249473727353148466, −20.03758206785540677354054227677, −19.57362819700897267802872230576, −18.40729279286542411496283386346, −17.34251807756756387574417685074, −16.680421811589320360980135792227, −15.86367566568131900284409114011, −15.58124282893195219303800269291, −14.094310061959056971947094066185, −12.649069652190304101053369786714, −12.25157062463682002590635158772, −11.02481738920672104554224918516, −10.400307626171246800524702620866, −9.68460077885569038591804330597, −8.71882843860539296597834806372, −7.87978949734332497535668019048, −6.93483344276089395026828871816, −5.50363759892589540467874746454, −4.27497542954247501643207265326, −3.58716989187106301067019279767, −2.50227783655047562826640070361, −0.58062677563459330181843686783, 0.46164012280288475353094728499, 2.3446122772746657828906715241, 2.81488693718672498802102538476, 4.88833851591345014692614867919, 5.94239118030189219156189207006, 6.84144958468685649925484831179, 7.48606956951288184698791230109, 8.19554030447409992293117320310, 9.32666838922822212766788816536, 10.26124953321356033852424625890, 11.44226826062326982776897930269, 11.87854735894773061712899239611, 13.03990530463796379267089717274, 14.06873350052404683464851446952, 15.15323357300877753460785028743, 15.79860298950657609717695552349, 16.555312271217322292624361227883, 17.86993822523750500085707069051, 18.18228270587855172299969406828, 19.05795676357533605720642932292, 19.631953824879740576650455185235, 20.2858375704271255225521094496, 22.02513453140381099406963271968, 22.666270184744859277996126740348, 23.681581340310064339496889904706

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