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Properties

Label 1-6025-6025.167-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.997 - 0.0671i$
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  + (0.629 − 0.777i)2-s + (0.998 − 0.0523i)3-s + (−0.207 − 0.978i)4-s + (0.587 − 0.809i)6-s + (0.566 − 0.824i)7-s + (−0.891 − 0.453i)8-s + (0.994 − 0.104i)9-s + (−0.284 + 0.958i)11-s + (−0.258 − 0.965i)12-s + (−0.608 + 0.793i)13-s + (−0.284 − 0.958i)14-s + (−0.913 + 0.406i)16-s + (−0.852 − 0.522i)17-s + (0.544 − 0.838i)18-s + (−0.566 + 0.824i)19-s + ⋯
L(s)  = 1  + (0.629 − 0.777i)2-s + (0.998 − 0.0523i)3-s + (−0.207 − 0.978i)4-s + (0.587 − 0.809i)6-s + (0.566 − 0.824i)7-s + (−0.891 − 0.453i)8-s + (0.994 − 0.104i)9-s + (−0.284 + 0.958i)11-s + (−0.258 − 0.965i)12-s + (−0.608 + 0.793i)13-s + (−0.284 − 0.958i)14-s + (−0.913 + 0.406i)16-s + (−0.852 − 0.522i)17-s + (0.544 − 0.838i)18-s + (−0.566 + 0.824i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.997 - 0.0671i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.997 - 0.0671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.148884510 - 0.1058383932i\)
\(L(\frac12)\) \(\approx\) \(3.148884510 - 0.1058383932i\)
\(L(1)\) \(\approx\) \(1.763851557 - 0.6622584025i\)
\(L(1)\) \(\approx\) \(1.763851557 - 0.6622584025i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.629 - 0.777i)T \)
3 \( 1 + (0.998 - 0.0523i)T \)
7 \( 1 + (0.566 - 0.824i)T \)
11 \( 1 + (-0.284 + 0.958i)T \)
13 \( 1 + (-0.608 + 0.793i)T \)
17 \( 1 + (-0.852 - 0.522i)T \)
19 \( 1 + (-0.566 + 0.824i)T \)
23 \( 1 + (0.760 + 0.649i)T \)
29 \( 1 + (0.933 + 0.358i)T \)
31 \( 1 + (0.477 + 0.878i)T \)
37 \( 1 + (-0.566 + 0.824i)T \)
41 \( 1 + (0.453 + 0.891i)T \)
43 \( 1 + (0.0784 + 0.996i)T \)
47 \( 1 + (0.987 - 0.156i)T \)
53 \( 1 + (-0.777 + 0.629i)T \)
59 \( 1 + (-0.838 + 0.544i)T \)
61 \( 1 + (-0.891 + 0.453i)T \)
67 \( 1 + (-0.629 + 0.777i)T \)
71 \( 1 + (0.688 - 0.725i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (0.156 - 0.987i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.477 - 0.878i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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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

−17.48955040229791960318564463823, −17.1677353644767867970723145768, −16.04316046186387411551695167172, −15.427300245739268348548798404170, −15.26146810226149250172980390793, −14.50155290119280341510049284922, −13.868998923649611934589471194849, −13.31817535497244079275695028733, −12.603428390033278513256983771292, −12.17240663157383660011239177666, −11.03745855202578207910013384209, −10.57728005320264287589516260618, −9.252100633445050717075283772553, −8.90736102611377599797121542398, −8.21739013788675138944969385825, −7.846526585986952149479643359153, −6.930364325157730332381303179311, −6.24542390451341412872174366408, −5.43412245570241687427553247466, −4.75196711882481202583696144968, −4.15186285441005689916730180152, −3.1899126753156139645470204092, −2.58343824095361250519447548155, −2.10423192788134376392253245679, −0.49572489347374562035520002168, 1.25407440020621871422747925148, 1.63323113206811515235761945063, 2.53479994169313241921762049916, 3.0856666787574782385283610148, 4.0868619111448494975492103098, 4.62533199033377424717827632101, 4.89626751287075495668689564494, 6.31968612565624391038040414195, 6.99856039507859548589352743899, 7.54100918866953342907854935697, 8.48439009781105807466022132433, 9.196094273166522765372090021203, 9.832769901785869769185775959465, 10.41039141021073595676626474201, 11.030824334020257985739003371628, 11.973918801282420912221785668290, 12.44573681868684001595969218781, 13.24370707248089944300887365954, 13.78120757000113992809134252787, 14.280935609492537285714243155782, 14.87936427990504280645515080855, 15.40492494247152087033645518715, 16.19205123742609733845976212849, 17.20672595499540881134668864958, 17.88750245030294704428023099250

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