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Properties

Label 1-6025-6025.4562-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.937 + 0.347i$
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  + (−0.453 + 0.891i)2-s + (−0.453 + 0.891i)3-s + (−0.587 − 0.809i)4-s + (−0.587 − 0.809i)6-s + (0.233 + 0.972i)7-s + (0.987 − 0.156i)8-s + (−0.587 − 0.809i)9-s + (−0.0784 + 0.996i)11-s + (0.987 − 0.156i)12-s + (0.233 − 0.972i)13-s + (−0.972 − 0.233i)14-s + (−0.309 + 0.951i)16-s + (−0.382 − 0.923i)17-s + (0.987 − 0.156i)18-s + (0.852 − 0.522i)19-s + ⋯
L(s)  = 1  + (−0.453 + 0.891i)2-s + (−0.453 + 0.891i)3-s + (−0.587 − 0.809i)4-s + (−0.587 − 0.809i)6-s + (0.233 + 0.972i)7-s + (0.987 − 0.156i)8-s + (−0.587 − 0.809i)9-s + (−0.0784 + 0.996i)11-s + (0.987 − 0.156i)12-s + (0.233 − 0.972i)13-s + (−0.972 − 0.233i)14-s + (−0.309 + 0.951i)16-s + (−0.382 − 0.923i)17-s + (0.987 − 0.156i)18-s + (0.852 − 0.522i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.937 + 0.347i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4562, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.937 + 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7450405080 + 0.1334941084i\)
\(L(\frac12)\) \(\approx\) \(0.7450405080 + 0.1334941084i\)
\(L(1)\) \(\approx\) \(0.5656032280 + 0.3857188184i\)
\(L(1)\) \(\approx\) \(0.5656032280 + 0.3857188184i\)

Euler product

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

−17.78626457345741684552379348706, −17.21998977975665039371908795956, −16.453241993649123743242280355917, −16.29852298499933134446576005982, −14.879123990205255828483696080, −13.80848626990935727207181013204, −13.66869094467591306927842477760, −13.18960829029512624995261502209, −12.152818596291343197092534277435, −11.64900761982413243513844880638, −11.22322420285572374806289691472, −10.467579743585089698298717934498, −9.97051173117854926139141054819, −8.87706398994774828766560864382, −8.3263949457625003366866214405, −7.74597047635870182338286927903, −6.98619692546235100485121722797, −6.360811039203336432945076905819, −5.4012626745140470485601473378, −4.602991474084344998506716906811, −3.74623994282005260505661852049, −3.158866149710577724452291561494, −2.08395445940671266356795058850, −1.39660706084191162390580533712, −0.877913792298988724014272394073, 0.31028445758439719657944050782, 1.36003646010233743764218123021, 2.5824616006876417221104982070, 3.27679255293640610647041413828, 4.609893497220719883006628302189, 4.898189834600554826165864270865, 5.39649375756966454832160666422, 6.31559515470489115481587491626, 6.82472922911080547826538500197, 7.74112667893857504952126671660, 8.57524264794343138542756822508, 8.969404259232009381993495993861, 9.820651311165402763086345483233, 10.161944655705843500137751097524, 10.98306790230763000626527902110, 11.74885352242663966734919769655, 12.3687473121857134817062316173, 13.285091620039851989369635780384, 14.07664701711598727135267111904, 14.9023097344027586123239776782, 15.25424806518074936839733803853, 15.784807569034319036135789345836, 16.22675424299306311532390742411, 17.23817536400289368770475326731, 17.556207562830573967295405675937

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