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Properties

Label 1-6025-6025.2548-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.0845 + 0.996i$
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 + (−0.972 − 0.233i)7-s + (−0.987 − 0.156i)8-s + (−0.587 + 0.809i)9-s + (−0.996 + 0.0784i)11-s + (−0.987 − 0.156i)12-s + (−0.972 + 0.233i)13-s + (−0.233 − 0.972i)14-s + (−0.309 − 0.951i)16-s + (−0.923 − 0.382i)17-s + (−0.987 − 0.156i)18-s + (0.522 − 0.852i)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 + (−0.972 − 0.233i)7-s + (−0.987 − 0.156i)8-s + (−0.587 + 0.809i)9-s + (−0.996 + 0.0784i)11-s + (−0.987 − 0.156i)12-s + (−0.972 + 0.233i)13-s + (−0.233 − 0.972i)14-s + (−0.309 − 0.951i)16-s + (−0.923 − 0.382i)17-s + (−0.987 − 0.156i)18-s + (0.522 − 0.852i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0845 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0845 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.0845 + 0.996i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (2548, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.0845 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6509424093 + 0.5980360954i\)
\(L(\frac12)\) \(\approx\) \(0.6509424093 + 0.5980360954i\)
\(L(1)\) \(\approx\) \(0.6525651276 + 0.6534885009i\)
\(L(1)\) \(\approx\) \(0.6525651276 + 0.6534885009i\)

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 \( 1 + (-0.972 - 0.233i)T \)
11 \( 1 + (-0.996 + 0.0784i)T \)
13 \( 1 + (-0.972 + 0.233i)T \)
17 \( 1 + (-0.923 - 0.382i)T \)
19 \( 1 + (0.522 - 0.852i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + (-0.382 + 0.923i)T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (0.891 - 0.453i)T \)
43 \( 1 + (-0.233 - 0.972i)T \)
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 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 + (-0.972 - 0.233i)T \)
79 \( 1 + (-0.987 - 0.156i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (-0.760 - 0.649i)T \)
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

−17.74074621660542720739730777498, −17.17524370527381241295369581031, −16.109578413607182190514799055317, −15.289155901189128124040902850583, −14.85877561794163703487857089072, −14.08665517024778593173732578256, −13.22450745830656784621983056204, −13.03451111999199865258815395326, −12.52046394796519236529282483662, −11.74036855145537880792071612578, −11.13281736616250341441387137666, −10.19669888734587491866456271908, −9.64996489556479209288792810130, −9.046160681251838159360082952063, −8.24912645583351714916096207291, −7.43774034767073002056150490281, −6.78202865733236441780152741857, −5.73852885067753661427187646708, −5.578007227352963201948194319269, −4.37466956517321247960987126780, −3.56901609420556887531273299820, −2.83651986138655372734696580160, −2.414702026594134187875693252599, −1.6390948612004830506412343569, −0.522982588894921627133292920154, 0.30088467609733592536052439479, 2.27363391937167537169637756173, 2.93401499211045835462234816423, 3.42360813331866349982987537515, 4.37264403820130882732624590119, 5.011660082005155373706250584662, 5.360285686557197871543381736591, 6.4720621102611850833127870176, 7.18447295868317057475877430152, 7.565385513001390754186076225045, 8.65849915500299651162217082903, 9.16626683713279357898181015569, 9.602574456044973558151144069275, 10.552247536756611264348752829705, 11.07563783252460229853513941119, 12.230444875414963927744386813880, 12.81282751797222568877187791630, 13.45677394618473225282022210158, 14.01614199551368894236172533835, 14.65815637177742705614582182790, 15.56446033380943541875037334158, 15.67498637929824390153453182578, 16.29409692180606825720342391803, 17.11027161088922097563758511925, 17.45758072787263361407710130695

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