Properties

Label 1-6025-6025.1833-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.517 + 0.855i$
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  + (−0.358 − 0.933i)2-s + (0.544 + 0.838i)3-s + (−0.743 + 0.669i)4-s + (0.587 − 0.809i)6-s + (0.430 + 0.902i)7-s + (0.891 + 0.453i)8-s + (−0.406 + 0.913i)9-s + (0.688 + 0.725i)11-s + (−0.965 − 0.258i)12-s + (0.991 + 0.130i)13-s + (0.688 − 0.725i)14-s + (0.104 − 0.994i)16-s + (−0.852 − 0.522i)17-s + (0.998 + 0.0523i)18-s + (0.430 + 0.902i)19-s + ⋯
L(s)  = 1  + (−0.358 − 0.933i)2-s + (0.544 + 0.838i)3-s + (−0.743 + 0.669i)4-s + (0.587 − 0.809i)6-s + (0.430 + 0.902i)7-s + (0.891 + 0.453i)8-s + (−0.406 + 0.913i)9-s + (0.688 + 0.725i)11-s + (−0.965 − 0.258i)12-s + (0.991 + 0.130i)13-s + (0.688 − 0.725i)14-s + (0.104 − 0.994i)16-s + (−0.852 − 0.522i)17-s + (0.998 + 0.0523i)18-s + (0.430 + 0.902i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.517 + 0.855i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.517 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8111882532 + 1.438878998i\)
\(L(\frac12)\) \(\approx\) \(0.8111882532 + 1.438878998i\)
\(L(1)\) \(\approx\) \(1.049685388 + 0.2625690366i\)
\(L(1)\) \(\approx\) \(1.049685388 + 0.2625690366i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.358 - 0.933i)T \)
3 \( 1 + (0.544 + 0.838i)T \)
7 \( 1 + (0.430 + 0.902i)T \)
11 \( 1 + (0.688 + 0.725i)T \)
13 \( 1 + (0.991 + 0.130i)T \)
17 \( 1 + (-0.852 - 0.522i)T \)
19 \( 1 + (0.430 + 0.902i)T \)
23 \( 1 + (0.760 + 0.649i)T \)
29 \( 1 + (-0.777 + 0.629i)T \)
31 \( 1 + (0.999 + 0.0261i)T \)
37 \( 1 + (-0.430 - 0.902i)T \)
41 \( 1 + (0.453 + 0.891i)T \)
43 \( 1 + (0.0784 + 0.996i)T \)
47 \( 1 + (-0.987 + 0.156i)T \)
53 \( 1 + (-0.933 - 0.358i)T \)
59 \( 1 + (-0.0523 - 0.998i)T \)
61 \( 1 + (-0.891 + 0.453i)T \)
67 \( 1 + (0.358 + 0.933i)T \)
71 \( 1 + (-0.284 - 0.958i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (0.156 - 0.987i)T \)
83 \( 1 + (0.978 - 0.207i)T \)
89 \( 1 + (-0.999 - 0.0261i)T \)
97 \( 1 + (-0.913 + 0.406i)T \)
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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

−17.378402426613285180587702662390, −17.07338073959784367923345589663, −16.25854938867299029913185593131, −15.3234924466666419144958502632, −15.03330959035431628606984134594, −13.98098995464963325708829470042, −13.73599647445833849245193022022, −13.32009444448757689876018362921, −12.457420706232232609306630167622, −11.28407677991254884874548117760, −10.99856682811279177896247544446, −10.031257825248640546698576934335, −9.0726415530548776625022696561, −8.70203203307264119592910127235, −8.099670209381991209908065381788, −7.42644797867951944880358532369, −6.61175692488717474399624273778, −6.43547975982038899803830382474, −5.48142272710320052332350448577, −4.481207666264333932534514450717, −3.8634744440425623944865349364, −3.03792628609642876266458031904, −1.80534398580004587459879385562, −1.1265389082654672330355721050, −0.46257218851467212766417315134, 1.39610615104019794479462991654, 1.82319040362048547265056079572, 2.78403857058181822941792990600, 3.35201426650702237926593887177, 4.141206224572939080718292078750, 4.75654466442906680431816166429, 5.4093235112713411943116675030, 6.42882785374992106192738925554, 7.55863276338403113462666657263, 8.16042115614865682276783256304, 8.896147465022336384736389819837, 9.31594290883976195160503432924, 9.776440240372720462745227499664, 10.729886997014437334853094083882, 11.333545571170732484591486887370, 11.68435683241920535782201510372, 12.64250667817014045856570538670, 13.24056308093413500190596941722, 14.077516280491036724818225554586, 14.56419557040933871042602096287, 15.2849409314668086176717372534, 16.04535827106585403486091188338, 16.53502720515394707106189380260, 17.58501509086876016161597759967, 17.87734828668389309443655390318

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