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Properties

Label 1-6025-6025.2077-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.382 - 0.924i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (0.998 + 0.0523i)2-s + (−0.0523 − 0.998i)3-s + (0.994 + 0.104i)4-s − i·6-s + (−0.902 − 0.430i)7-s + (0.987 + 0.156i)8-s + (−0.994 + 0.104i)9-s + (0.430 − 0.902i)11-s + (0.0523 − 0.998i)12-s + (0.983 + 0.182i)13-s + (−0.878 − 0.477i)14-s + (0.978 + 0.207i)16-s + (−0.0784 + 0.996i)17-s + (−0.998 + 0.0523i)18-s + (0.477 + 0.878i)19-s + ⋯
L(s)  = 1  + (0.998 + 0.0523i)2-s + (−0.0523 − 0.998i)3-s + (0.994 + 0.104i)4-s − i·6-s + (−0.902 − 0.430i)7-s + (0.987 + 0.156i)8-s + (−0.994 + 0.104i)9-s + (0.430 − 0.902i)11-s + (0.0523 − 0.998i)12-s + (0.983 + 0.182i)13-s + (−0.878 − 0.477i)14-s + (0.978 + 0.207i)16-s + (−0.0784 + 0.996i)17-s + (−0.998 + 0.0523i)18-s + (0.477 + 0.878i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.382 - 0.924i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (2077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.382 - 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.096159043 - 2.069727129i\)
\(L(\frac12)\) \(\approx\) \(3.096159043 - 2.069727129i\)
\(L(1)\) \(\approx\) \(1.859842206 - 0.6635811184i\)
\(L(1)\) \(\approx\) \(1.859842206 - 0.6635811184i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.998 + 0.0523i)T \)
3 \( 1 + (-0.0523 - 0.998i)T \)
7 \( 1 + (-0.902 - 0.430i)T \)
11 \( 1 + (0.430 - 0.902i)T \)
13 \( 1 + (0.983 + 0.182i)T \)
17 \( 1 + (-0.0784 + 0.996i)T \)
19 \( 1 + (0.477 + 0.878i)T \)
23 \( 1 + (0.522 - 0.852i)T \)
29 \( 1 + (-0.998 - 0.0523i)T \)
31 \( 1 + (0.566 - 0.824i)T \)
37 \( 1 + (0.688 - 0.725i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (0.996 + 0.0784i)T \)
47 \( 1 + (-0.891 + 0.453i)T \)
53 \( 1 + (-0.544 + 0.838i)T \)
59 \( 1 + (-0.933 - 0.358i)T \)
61 \( 1 + (-0.156 + 0.987i)T \)
67 \( 1 + (0.838 - 0.544i)T \)
71 \( 1 + (0.942 - 0.333i)T \)
73 \( 1 + (-0.649 - 0.760i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (0.942 - 0.333i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−17.613801959782315975740010297631, −16.99860975599319545576356236452, −16.03524518026333054799237152692, −15.87113395015908813535442421889, −15.30702245750911241633635124563, −14.65763099035022139929798171742, −13.885977396300454772072642108759, −13.30850048449728372615481676481, −12.62872889755474578774820016225, −11.84154013804865819232233972363, −11.311780394690039221470287635553, −10.75309615467276376092411335451, −9.70117103796507193124212991828, −9.51371072079465386552820479627, −8.663946222687129867515233372059, −7.56000433072845063601624486314, −6.81661024698162101221622332623, −6.21313754414201481106786415012, −5.3968934396330099585355078951, −4.93770157243237899655969619283, −4.123503732249072158289975684163, −3.40396130029214031531751779913, −2.95540180992562188537079152155, −2.11180456213278168762463452548, −0.90536359818459928179594673465, 0.80075125191348180762611341109, 1.460612926245647003574409823663, 2.40373104499174659831104509157, 3.24750866146155258646827569088, 3.705212208240315888965380211676, 4.49377621705407581218503233331, 5.77661268070701700627097604266, 6.172920895313693516059439918015, 6.37818469366443985440210724243, 7.45217301459522232434555880336, 7.91056438030882889647013496674, 8.75132166246754899508883388880, 9.6008724029054518620038124254, 10.80206553095368316761499965058, 10.99951663012814201640322055099, 11.822072538473005524449609260024, 12.62482100114854002545579600028, 12.97562823672442330733165483924, 13.55384264134357662024827003608, 14.17953378014824388131457838217, 14.671580508434502322369396480704, 15.61440445219073381158126187209, 16.44150466200618883755994529436, 16.67555094960322802691616840073, 17.386396673186166018606099879115

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