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Properties

Label 1-39e2-1521.994-r1-0-0
Degree $1$
Conductor $1521$
Sign $0.566 - 0.824i$
Analytic cond. $163.454$
Root an. cond. $163.454$
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  + (0.822 − 0.568i)2-s + (0.354 − 0.935i)4-s + (0.316 − 0.948i)5-s + (−0.721 + 0.692i)7-s + (−0.239 − 0.970i)8-s + (−0.278 − 0.960i)10-s + (0.822 + 0.568i)11-s + (−0.200 + 0.979i)14-s + (−0.748 − 0.663i)16-s + (−0.278 + 0.960i)17-s + (0.866 + 0.5i)19-s + (−0.774 − 0.632i)20-s + 22-s + (0.5 − 0.866i)23-s + (−0.799 − 0.600i)25-s + ⋯
L(s)  = 1  + (0.822 − 0.568i)2-s + (0.354 − 0.935i)4-s + (0.316 − 0.948i)5-s + (−0.721 + 0.692i)7-s + (−0.239 − 0.970i)8-s + (−0.278 − 0.960i)10-s + (0.822 + 0.568i)11-s + (−0.200 + 0.979i)14-s + (−0.748 − 0.663i)16-s + (−0.278 + 0.960i)17-s + (0.866 + 0.5i)19-s + (−0.774 − 0.632i)20-s + 22-s + (0.5 − 0.866i)23-s + (−0.799 − 0.600i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.566 - 0.824i$
Analytic conductor: \(163.454\)
Root analytic conductor: \(163.454\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (994, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1521,\ (1:\ ),\ 0.566 - 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.489796836 - 1.836185979i\)
\(L(\frac12)\) \(\approx\) \(3.489796836 - 1.836185979i\)
\(L(1)\) \(\approx\) \(1.653763655 - 0.7188293488i\)
\(L(1)\) \(\approx\) \(1.653763655 - 0.7188293488i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.822 - 0.568i)T \)
5 \( 1 + (0.316 - 0.948i)T \)
7 \( 1 + (-0.721 + 0.692i)T \)
11 \( 1 + (0.822 + 0.568i)T \)
17 \( 1 + (-0.278 + 0.960i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.568 + 0.822i)T \)
31 \( 1 + (-0.600 - 0.799i)T \)
37 \( 1 + (0.600 + 0.799i)T \)
41 \( 1 + (0.534 + 0.845i)T \)
43 \( 1 + (0.919 - 0.391i)T \)
47 \( 1 + (-0.160 + 0.987i)T \)
53 \( 1 + (-0.970 + 0.239i)T \)
59 \( 1 + (-0.663 - 0.748i)T \)
61 \( 1 + (0.278 + 0.960i)T \)
67 \( 1 + (0.160 - 0.987i)T \)
71 \( 1 + (0.534 + 0.845i)T \)
73 \( 1 + (0.822 + 0.568i)T \)
79 \( 1 + (0.987 + 0.160i)T \)
83 \( 1 + (0.999 + 0.0402i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.979 + 0.200i)T \)
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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

−20.66807369725449072742851949569, −19.75884231659658907254452767291, −19.19771709929597624135339419907, −17.98848584794753006769242486812, −17.50750352980814562328713372869, −16.58755903798854726586597942322, −15.950678508406992433727070422829, −15.261122123515470916022971249212, −14.20938692580674091038988394233, −13.89159260741509034331557724604, −13.28632612921909723716864432018, −12.25707288278801896560487209792, −11.34698160767492927942426421723, −10.876500641860207767860374059774, −9.63278214778633392994496153816, −9.05630538503629734370053095879, −7.63738223231591877408061789857, −7.14055526177506176740526370679, −6.45718400866186668458299605388, −5.76050675033647860481551225951, −4.7403770933258366203718623215, −3.61400359507816373126680638591, −3.23530169654626048179683473045, −2.2411003427255850447784921250, −0.69244124693366492694434578372, 0.82509012573926987028764081468, 1.66016130866454655336902185457, 2.56927931085504194182352267188, 3.590502270691325039280386925882, 4.43143235580299541326369635380, 5.187024820994910646899163485772, 6.1142368493772994951548264240, 6.55270517312959580882437201058, 7.92260760232029759973051496465, 9.135450585996786003960284090253, 9.437238412870947182474745372204, 10.336495478662375475009200479266, 11.34595918509419870914538842854, 12.26285349463219699150021586688, 12.59460607949279731180483174411, 13.22083602169155939857337970020, 14.22398420604591334660589822193, 14.85839701198584965357736015296, 15.73354127347677428349724880398, 16.39140837865007080612738952002, 17.19704087916772276985249942022, 18.23711193956821238657235623678, 19.00695895601326341243211374157, 19.83966540786471555202672275130, 20.26090878780220254097132060378

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