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Properties

Label 1-6025-6025.211-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.664 - 0.747i$
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  + (−0.951 − 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)6-s + (−0.707 − 0.707i)7-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (−0.891 + 0.453i)11-s + (−0.951 + 0.309i)12-s + (−0.891 − 0.453i)13-s + (0.453 + 0.891i)14-s + (0.309 + 0.951i)16-s + (0.156 − 0.987i)17-s + i·18-s + (−0.987 − 0.156i)19-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)6-s + (−0.707 − 0.707i)7-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (−0.891 + 0.453i)11-s + (−0.951 + 0.309i)12-s + (−0.891 − 0.453i)13-s + (0.453 + 0.891i)14-s + (0.309 + 0.951i)16-s + (0.156 − 0.987i)17-s + i·18-s + (−0.987 − 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.664 - 0.747i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.664 - 0.747i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1552661193 - 0.3457544430i\)
\(L(\frac12)\) \(\approx\) \(0.1552661193 - 0.3457544430i\)
\(L(1)\) \(\approx\) \(0.4439983866 - 0.06066367893i\)
\(L(1)\) \(\approx\) \(0.4439983866 - 0.06066367893i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.891 + 0.453i)T \)
13 \( 1 + (-0.891 - 0.453i)T \)
17 \( 1 + (0.156 - 0.987i)T \)
19 \( 1 + (-0.987 - 0.156i)T \)
23 \( 1 + (-0.891 + 0.453i)T \)
29 \( 1 + (0.587 - 0.809i)T \)
31 \( 1 + (0.156 - 0.987i)T \)
37 \( 1 + (0.453 - 0.891i)T \)
41 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (-0.587 + 0.809i)T \)
59 \( 1 + (0.951 - 0.309i)T \)
61 \( 1 + (0.951 + 0.309i)T \)
67 \( 1 + (0.587 + 0.809i)T \)
71 \( 1 + (0.987 - 0.156i)T \)
73 \( 1 + (-0.891 + 0.453i)T \)
79 \( 1 + (-0.587 + 0.809i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (0.453 + 0.891i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−17.88007964750921432417583945987, −17.45926308936169801662398349844, −16.63129854448210848773953625871, −16.19292600423582831352243223369, −15.66252372802269614463820295674, −14.658383555744761425798318824767, −14.247978530424269476481585399624, −13.090992440761329431971230354971, −12.548185627585790904817705724994, −12.09580850004501858615919345564, −11.24605994884754831344164512026, −10.55345356776553144343118491512, −10.09437219701676131259239803185, −9.22576504340025882517364514986, −8.30512649190294677401733419134, −8.123937096174111595906485874697, −7.15967153100001242761667393702, −6.44504043154531454659884672102, −6.09336045459859925696698607506, −5.37048554580113184891705964655, −4.56214597300067600253398999617, −3.101778562993439226340027063270, −2.410383080092472299961884239611, −1.86569826165123171210830994194, −0.79957427481052847014111053416, 0.25726356343266656926102724483, 0.73934424762203646541512061476, 2.401795516674138701714164428021, 2.62234940448129118197477920600, 3.846461221095558738478431136130, 4.223144539606838086308748140702, 5.303770212135218045565265330750, 5.99283524741757484598435248097, 6.83384564647595831725934708668, 7.46509835483350343571025271661, 8.05918429447548258504840370555, 9.12845841871351206741429651614, 9.68935587944391205582096542274, 10.134841034630350998128129387574, 10.609164560050009348179527140080, 11.325612067171461727003212721703, 12.07905497542266586784796930819, 12.630574038033357278031273310498, 13.31956159285953215746487244296, 14.35983949082755160986870226879, 15.19652773269197767663559860778, 15.85708672464107345363393389479, 16.09708813856035824028710489173, 17.01058237968642814972689117697, 17.40133471266137503970125257372

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