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Properties

Label 1-6025-6025.759-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.0968 + 0.995i$
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  + (−0.309 − 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)6-s + (−0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.309 − 0.951i)11-s − 12-s + 13-s + (−0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (0.309 + 0.951i)17-s + (0.809 − 0.587i)18-s + (0.809 + 0.587i)19-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)6-s + (−0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.309 − 0.951i)11-s − 12-s + 13-s + (−0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (0.309 + 0.951i)17-s + (0.809 − 0.587i)18-s + (0.809 + 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0968 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0968 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.0968 + 0.995i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.0968 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5098024576 + 0.5618279168i\)
\(L(\frac12)\) \(\approx\) \(0.5098024576 + 0.5618279168i\)
\(L(1)\) \(\approx\) \(0.9047847592 - 0.1459396914i\)
\(L(1)\) \(\approx\) \(0.9047847592 - 0.1459396914i\)

Euler product

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

−17.94113798361675171364869573131, −16.73620579935325563609646575412, −16.02254624561444153237732899918, −15.62564563834327537624001690776, −15.051519490024638556775620058098, −14.275316664769362298419487619674, −13.62137189159209954297281335681, −13.2765388278388993169464498314, −12.320065144595560484326570417094, −11.98762232309618870533385262180, −10.59060492862448692741191063846, −9.9394357107042166126359781917, −9.32795079740886570659589854494, −8.74615667674529984835385256744, −8.23288826221832765165068126375, −7.32598275905037557615349743342, −6.848657486477747954771910616249, −6.37069908176541101121117750846, −5.36132380451308937531673257590, −4.82328280621085651861789782242, −3.61291495893890441836980116555, −3.16132397965716724649992142533, −2.07982318921188170075471784361, −1.332956676361918480274062489809, −0.19747446781488897930787300025, 1.11908720571632914248811701137, 1.81455791421995251226816003269, 2.87195534616259216015702808074, 3.49448862833538306363858461641, 3.74759546848543315883602300217, 4.56226221019497284469218907615, 5.64672197310115643279190191569, 6.26906713782956148838795499810, 7.669611333170527575416769494009, 7.94557914557286207269208482759, 8.65540192658241361990680283708, 9.43862913542368915099418461083, 9.89972817635095501261159696257, 10.54331817803024172484145039679, 11.03994378674719933698557473800, 11.83642399516335827199906335206, 12.764179440173389658702260138618, 13.36572996512937241899180827796, 13.79923336970176656975158431137, 14.27866546797965148783617832220, 15.423442586047826840260320993024, 15.96621264333518200451259036723, 16.63172323959730488795974399136, 17.08415901610128624689999191785, 18.2091371863809742155020302427

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