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Properties

Label 1-539-539.153-r0-0-0
Degree $1$
Conductor $539$
Sign $-0.938 + 0.345i$
Analytic cond. $2.50310$
Root an. cond. $2.50310$
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  + (0.222 + 0.974i)2-s + (−0.623 − 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (0.623 − 0.781i)6-s + (−0.623 − 0.781i)8-s + (−0.222 + 0.974i)9-s + (0.623 − 0.781i)10-s + (0.900 + 0.433i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s − 18-s + 19-s + (0.900 + 0.433i)20-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)2-s + (−0.623 − 0.781i)3-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (0.623 − 0.781i)6-s + (−0.623 − 0.781i)8-s + (−0.222 + 0.974i)9-s + (0.623 − 0.781i)10-s + (0.900 + 0.433i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s − 18-s + 19-s + (0.900 + 0.433i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.938 + 0.345i$
Analytic conductor: \(2.50310\)
Root analytic conductor: \(2.50310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 539,\ (0:\ ),\ -0.938 + 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03084826477 + 0.1731454482i\)
\(L(\frac12)\) \(\approx\) \(0.03084826477 + 0.1731454482i\)
\(L(1)\) \(\approx\) \(0.5928733855 + 0.1035503454i\)
\(L(1)\) \(\approx\) \(0.5928733855 + 0.1035503454i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.222 + 0.974i)T \)
3 \( 1 + (-0.623 - 0.781i)T \)
5 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + (-0.623 + 0.781i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 - T \)
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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

−22.765621010362860358465742609735, −22.056408567001243197103616048107, −21.687505308963163398435791764568, −20.53870117721310319829712778784, −19.867182371896781741666490385056, −18.916799623522356761620784296474, −18.13021250814784495941347165913, −17.350930656247184400544591243400, −16.15290293174861672678437247769, −15.36262933995487520150426397259, −14.459264466021133338900782458247, −13.73665463434069203792637482805, −12.24993376093448007815323341333, −11.78696440186599854589710470311, −10.95701310823334762538361840282, −10.30833229377656697771595202596, −9.41897165814804155393329749781, −8.45137801272409606312303557849, −6.95562525472208686340269175579, −5.97189356527757410553604219774, −4.778045284580301966177102995607, −4.027968612495926457448474293615, −3.22906600929264967541154040942, −1.959706551956368609388156238269, −0.104136316640896682715530276946, 1.2078138332597902499395840553, 3.057962757497157282997925904101, 4.43632619365159572566304844605, 5.21896444238399269018850848236, 5.99717748685895694668904432172, 7.17487759663619088618780321087, 7.76380739660639119015404213335, 8.57681581844388082308075045630, 9.666831889244968599569983777332, 11.08993095133642243665860250763, 12.098264854876750843413629015656, 12.71024060984719116662655699145, 13.47292523377329896489679112254, 14.36834721198877860605946828830, 15.753598557649104776412360531794, 15.993280952508197654230811220059, 17.055268559377644933914284263015, 17.79483658547536526768013203107, 18.36747093313349665641692499803, 19.60871042424616187135192600490, 20.230920313200695101862838721473, 21.67129843919772399949664780087, 22.480924442030811299387756683929, 23.14023501086004672187690111655, 23.88073235597852639207383397204

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