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Properties

Label 1-6025-6025.5519-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.708 + 0.705i$
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  + (0.104 + 0.994i)2-s + (−0.913 − 0.406i)3-s + (−0.978 + 0.207i)4-s + (0.309 − 0.951i)6-s + (0.669 + 0.743i)7-s + (−0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.104 + 0.994i)11-s + (0.978 + 0.207i)12-s + (0.669 − 0.743i)13-s + (−0.669 + 0.743i)14-s + (0.913 − 0.406i)16-s + 17-s + (−0.669 + 0.743i)18-s + (0.978 + 0.207i)19-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.913 − 0.406i)3-s + (−0.978 + 0.207i)4-s + (0.309 − 0.951i)6-s + (0.669 + 0.743i)7-s + (−0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.104 + 0.994i)11-s + (0.978 + 0.207i)12-s + (0.669 − 0.743i)13-s + (−0.669 + 0.743i)14-s + (0.913 − 0.406i)16-s + 17-s + (−0.669 + 0.743i)18-s + (0.978 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.708 + 0.705i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (5519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.708 + 0.705i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5904802043 + 1.428988819i\)
\(L(\frac12)\) \(\approx\) \(0.5904802043 + 1.428988819i\)
\(L(1)\) \(\approx\) \(0.7859081635 + 0.5451628150i\)
\(L(1)\) \(\approx\) \(0.7859081635 + 0.5451628150i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.104 + 0.994i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
7 \( 1 + (0.669 + 0.743i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (0.669 - 0.743i)T \)
17 \( 1 + T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 - T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (0.104 + 0.994i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−17.33534141960911175474026893551, −17.08618859067530315468562250724, −16.26010387004733898293256761950, −15.65868743393434398632734986195, −14.54677769232914296654182116646, −14.12222998569813347787592819476, −13.45986102197188340430194426776, −12.78302837068991072545213520615, −11.798236254117550816645280500600, −11.514091201370261386193558116271, −11.00428143426863004287095661990, −10.32227220876296399859835829632, −9.778547179243040650904583845328, −8.93075262333962610480427689284, −8.340501755330480854887166362535, −7.32643690542306133314921275916, −6.57727104261151587681495414347, −5.47810474651922653828931087751, −5.29151084680486438633058910806, −4.35974725625483727074999878918, −3.652166581219889575390614879981, −3.26712899395161965160502467084, −1.887436267503796784140191382871, −1.112614600699229402010185598110, −0.589513275003243930882010038580, 1.014712735179441707423249306065, 1.509539271046247691695385762487, 2.828830462551393425748513096268, 3.71451878090791254482144596362, 4.88798917027414054590109437676, 5.043992517424856348265073603953, 5.77433424719931165835624229433, 6.40938855732052630851166579545, 7.19143059715908758932556083123, 7.81824786832479453972319717511, 8.25496464267069743176060246285, 9.29161588871910971173621299058, 9.87450771364723781675734397018, 10.68074521833604889474029479930, 11.55608756624850227792189169788, 12.14180678839308867966090135067, 12.71830258525969304505217788365, 13.32859847469645884307647117161, 14.09059546216419190437649143517, 14.95859668086924342896108565389, 15.27883489494491180426944458138, 16.01424536713707400871158441649, 16.786543510055624983062475703429, 17.17346695681593952036192088297, 18.065270733960851452514201434316

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