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Properties

Label 1-6025-6025.567-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.501 - 0.865i$
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  + (0.156 − 0.987i)2-s + (−0.707 + 0.707i)3-s + (−0.951 − 0.309i)4-s + (0.587 + 0.809i)6-s + (−0.233 + 0.972i)7-s + (−0.453 + 0.891i)8-s − i·9-s + (−0.522 + 0.852i)11-s + (0.891 − 0.453i)12-s + (0.760 + 0.649i)13-s + (0.923 + 0.382i)14-s + (0.809 + 0.587i)16-s + (−0.760 − 0.649i)17-s + (−0.987 − 0.156i)18-s + (0.760 − 0.649i)19-s + ⋯
L(s)  = 1  + (0.156 − 0.987i)2-s + (−0.707 + 0.707i)3-s + (−0.951 − 0.309i)4-s + (0.587 + 0.809i)6-s + (−0.233 + 0.972i)7-s + (−0.453 + 0.891i)8-s − i·9-s + (−0.522 + 0.852i)11-s + (0.891 − 0.453i)12-s + (0.760 + 0.649i)13-s + (0.923 + 0.382i)14-s + (0.809 + 0.587i)16-s + (−0.760 − 0.649i)17-s + (−0.987 − 0.156i)18-s + (0.760 − 0.649i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.501 - 0.865i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.501 - 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8178355484 - 0.4713841158i\)
\(L(\frac12)\) \(\approx\) \(0.8178355484 - 0.4713841158i\)
\(L(1)\) \(\approx\) \(0.7566461017 - 0.1392488226i\)
\(L(1)\) \(\approx\) \(0.7566461017 - 0.1392488226i\)

Euler product

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

−17.65055413265536412336362599575, −17.22280992047308530781708547770, −16.39825130952701806779542583970, −16.04114670415666968317982320641, −15.49128386688567121860956501379, −14.24872481617141022034828685594, −13.95304515023241965987797739696, −13.19409289979853322906589546199, −12.89911222936134301607442012458, −12.070423252155594858163761370523, −11.19321421778199194913351339201, −10.413419471599433623540715021601, −10.10770609833157155557956980108, −8.76799220523265853196492811833, −8.17197463071489698568502786005, −7.78235539084815208331634385503, −6.87868341820282534507488890821, −6.3187734305816168479508215885, −5.90115356185702946858974462761, −5.03851349611627792355840439122, −4.369787411532453162181174859705, −3.515341329785175890182843837472, −2.72925151921467442248177560907, −1.26087459380043127409193432373, −0.715317912046745906645757275571, 0.38950926555802065194749328577, 1.54105200035040771052065881354, 2.364702705212940199850411137403, 3.10503181567202379737875195475, 3.86145262921908922815206195223, 4.740388333008419445091045849887, 5.056336072649639548845344058078, 5.88858014554928664892672100061, 6.53374585231630464239865535021, 7.562811985524134470868873696783, 8.67360218131878821992839912763, 9.29018291744327155974935197750, 9.56818690928094666391800047923, 10.40114619094679601647643211505, 11.1914888273706286210252022918, 11.49638772061423406804711830628, 12.27870984055317174443798190595, 12.67033952077678268106262634949, 13.63579257670284666776940430004, 14.149914442241125322382468005993, 15.1627277924298278517988805812, 15.76764487253607269870046558007, 15.93846071063309326619236267220, 17.214269548206135309140149231660, 17.85794090565677084501345038338

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