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Properties

Label 1-6025-6025.3321-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.408 - 0.912i$
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  + (−0.406 − 0.913i)2-s + (0.406 − 0.913i)3-s + (−0.669 + 0.743i)4-s − 6-s + (0.933 + 0.358i)7-s + (0.951 + 0.309i)8-s + (−0.669 − 0.743i)9-s + (0.933 + 0.358i)11-s + (0.406 + 0.913i)12-s + (0.629 + 0.777i)13-s + (−0.0523 − 0.998i)14-s + (−0.104 − 0.994i)16-s + (−0.156 + 0.987i)17-s + (−0.406 + 0.913i)18-s + (0.0523 − 0.998i)19-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (0.406 − 0.913i)3-s + (−0.669 + 0.743i)4-s − 6-s + (0.933 + 0.358i)7-s + (0.951 + 0.309i)8-s + (−0.669 − 0.743i)9-s + (0.933 + 0.358i)11-s + (0.406 + 0.913i)12-s + (0.629 + 0.777i)13-s + (−0.0523 − 0.998i)14-s + (−0.104 − 0.994i)16-s + (−0.156 + 0.987i)17-s + (−0.406 + 0.913i)18-s + (0.0523 − 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.408 - 0.912i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (3321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.408 - 0.912i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.110202004 - 1.713794232i\)
\(L(\frac12)\) \(\approx\) \(1.110202004 - 1.713794232i\)
\(L(1)\) \(\approx\) \(0.9314352491 - 0.6926278776i\)
\(L(1)\) \(\approx\) \(0.9314352491 - 0.6926278776i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.406 - 0.913i)T \)
3 \( 1 + (0.406 - 0.913i)T \)
7 \( 1 + (0.933 + 0.358i)T \)
11 \( 1 + (0.933 + 0.358i)T \)
13 \( 1 + (0.629 + 0.777i)T \)
17 \( 1 + (-0.156 + 0.987i)T \)
19 \( 1 + (0.0523 - 0.998i)T \)
23 \( 1 + (0.891 - 0.453i)T \)
29 \( 1 + (0.406 + 0.913i)T \)
31 \( 1 + (-0.777 - 0.629i)T \)
37 \( 1 + (-0.544 - 0.838i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.156 - 0.987i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.994 + 0.104i)T \)
59 \( 1 + (-0.207 + 0.978i)T \)
61 \( 1 + (0.951 + 0.309i)T \)
67 \( 1 + (-0.994 + 0.104i)T \)
71 \( 1 + (0.358 - 0.933i)T \)
73 \( 1 + (-0.987 - 0.156i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (0.358 - 0.933i)T \)
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

−17.679550301915569378259168263231, −17.18377986556478210531930884333, −16.53277789121722280801069361883, −15.992272693432393938706069829210, −15.31835067922531104373543463222, −14.72043740230178280636226460140, −14.182442090229171286232252001805, −13.74511769933964133906111478870, −12.96873444538949450244930967281, −11.64458716585684569714101046356, −11.17739524122727379043502648963, −10.45678423736084237212245856442, −9.81787238610929465130174816987, −9.15663054063587068216303974214, −8.49818732695071596498893036657, −8.02919653480604690570027080056, −7.34209796137067475741525955665, −6.450334186106459437358438280601, −5.61336654241766687882237600574, −5.08845011240512407083690670611, −4.3421211871444122148945942882, −3.72550861227009699651758171821, −2.86348323828932446307474175713, −1.5658372384419746029437643518, −0.923389627356180538262502488575, 0.6943418760043698780465291743, 1.556942760464314082177636138989, 1.93982434785690927653678330452, 2.681644388183285641857203615117, 3.71278047734381136132543796493, 4.18845156934066763398548411093, 5.159926904039623106789862998652, 6.06071318978211312929420867185, 7.19357092630685065569802430780, 7.24557307173227945872198879651, 8.554476644447317367733341352504, 8.87019321910204040344945707244, 9.0726687393761467885733764292, 10.40436768297907816985194758519, 11.00399315077156891630718247863, 11.652185019965873846076335274051, 12.13166920384167930681078012359, 12.771757572649881985078421330392, 13.47755540396536108287345078488, 14.07714432815794639830894524807, 14.68956554818777372455270318812, 15.30949760880656680517883343520, 16.55851438354402636496868239194, 17.14738466720353215642072640420, 17.73481520772317918862939677278

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