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Properties

Label 1-1205-1205.103-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.453 - 0.891i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + (−0.707 + 0.707i)2-s + (−0.891 − 0.453i)3-s − i·4-s + (0.951 − 0.309i)6-s + (0.972 + 0.233i)7-s + (0.707 + 0.707i)8-s + (0.587 + 0.809i)9-s + (0.382 + 0.923i)11-s + (−0.453 + 0.891i)12-s + (0.0784 − 0.996i)13-s + (−0.852 + 0.522i)14-s − 16-s + (−0.522 − 0.852i)17-s + (−0.987 − 0.156i)18-s + (0.923 − 0.382i)19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.891 − 0.453i)3-s − i·4-s + (0.951 − 0.309i)6-s + (0.972 + 0.233i)7-s + (0.707 + 0.707i)8-s + (0.587 + 0.809i)9-s + (0.382 + 0.923i)11-s + (−0.453 + 0.891i)12-s + (0.0784 − 0.996i)13-s + (−0.852 + 0.522i)14-s − 16-s + (−0.522 − 0.852i)17-s + (−0.987 − 0.156i)18-s + (0.923 − 0.382i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.453 - 0.891i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.453 - 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5914267093 - 0.3624749257i\)
\(L(\frac12)\) \(\approx\) \(0.5914267093 - 0.3624749257i\)
\(L(1)\) \(\approx\) \(0.6313843677 + 0.0006034111837i\)
\(L(1)\) \(\approx\) \(0.6313843677 + 0.0006034111837i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-0.891 - 0.453i)T \)
7 \( 1 + (0.972 + 0.233i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + (0.0784 - 0.996i)T \)
17 \( 1 + (-0.522 - 0.852i)T \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (-0.649 - 0.760i)T \)
29 \( 1 + (0.156 - 0.987i)T \)
31 \( 1 + (-0.852 + 0.522i)T \)
37 \( 1 + (-0.0784 - 0.996i)T \)
41 \( 1 + (-0.156 - 0.987i)T \)
43 \( 1 + (0.233 + 0.972i)T \)
47 \( 1 + (-0.987 - 0.156i)T \)
53 \( 1 + (0.156 - 0.987i)T \)
59 \( 1 + (-0.891 + 0.453i)T \)
61 \( 1 + (0.891 + 0.453i)T \)
67 \( 1 + (-0.987 + 0.156i)T \)
71 \( 1 + (-0.233 - 0.972i)T \)
73 \( 1 + (0.0784 + 0.996i)T \)
79 \( 1 + (0.891 - 0.453i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (-0.382 + 0.923i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−21.46050271583609697134775890287, −20.55341901406600852874919371229, −19.86866834751846693668431653594, −18.82928651962485715022555338180, −18.21145313912434319703044649036, −17.53296129358687156585233514805, −16.74563873518785503458440379536, −16.36356038444844038160198803968, −15.3509769956252581750346520291, −14.23836551273091115120513328433, −13.44825959448700779862503210615, −12.321303380389853772937475279149, −11.56965185333142069904995790846, −11.21217214300292248798285345545, −10.46793068966896643209371453889, −9.55861435219184487581394227630, −8.81642183467041273251855108944, −7.93861937905155682671004124184, −6.96883685169686320130601664739, −6.03478583792034918358518935309, −4.941526044718036023196859727993, −4.051411337333251168052570342705, −3.399104305318821633701922481429, −1.76862989629321339488682668145, −1.17861669899107187157158096690, 0.457629614539551982207976407291, 1.54822378488287060638794151601, 2.38978698482540074681579148673, 4.38208489057150448665211546931, 5.101883463606807754773511551112, 5.73640850785222229941775835173, 6.75593114940920305204538809319, 7.449117987728621235867181659917, 8.05444928177496601135494988653, 9.10600450380286778946561250128, 10.0109519159832832397654432068, 10.8026089009233289380780589214, 11.53175934049670254952186888029, 12.270301483229541529527600542663, 13.34920119306730928874615037915, 14.22095652872589823214318618654, 15.03105247790742213598280231224, 15.82677890447684212568528270123, 16.472804629554670092290008032749, 17.543194511233254249681587879497, 17.93076708306700227312156340931, 18.16362425936597105915962689258, 19.35034893887046472501111360946, 20.12418034148219470651938602093, 20.848335275620562666451278203611

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