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Properties

Label 1-6025-6025.5438-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.999 + 0.00416i$
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  + (0.891 + 0.453i)2-s + (0.891 + 0.453i)3-s + (0.587 + 0.809i)4-s + (0.587 + 0.809i)6-s + (−0.852 − 0.522i)7-s + (0.156 + 0.987i)8-s + (0.587 + 0.809i)9-s + (0.649 + 0.760i)11-s + (0.156 + 0.987i)12-s + (−0.852 + 0.522i)13-s + (−0.522 − 0.852i)14-s + (−0.309 + 0.951i)16-s + (−0.382 + 0.923i)17-s + (0.156 + 0.987i)18-s + (−0.233 + 0.972i)19-s + ⋯
L(s)  = 1  + (0.891 + 0.453i)2-s + (0.891 + 0.453i)3-s + (0.587 + 0.809i)4-s + (0.587 + 0.809i)6-s + (−0.852 − 0.522i)7-s + (0.156 + 0.987i)8-s + (0.587 + 0.809i)9-s + (0.649 + 0.760i)11-s + (0.156 + 0.987i)12-s + (−0.852 + 0.522i)13-s + (−0.522 − 0.852i)14-s + (−0.309 + 0.951i)16-s + (−0.382 + 0.923i)17-s + (0.156 + 0.987i)18-s + (−0.233 + 0.972i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.999 + 0.00416i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (5438, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.999 + 0.00416i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.007313426111 + 3.513204436i\)
\(L(\frac12)\) \(\approx\) \(0.007313426111 + 3.513204436i\)
\(L(1)\) \(\approx\) \(1.564062649 + 1.371168739i\)
\(L(1)\) \(\approx\) \(1.564062649 + 1.371168739i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.891 + 0.453i)T \)
3 \( 1 + (0.891 + 0.453i)T \)
7 \( 1 + (-0.852 - 0.522i)T \)
11 \( 1 + (0.649 + 0.760i)T \)
13 \( 1 + (-0.852 + 0.522i)T \)
17 \( 1 + (-0.382 + 0.923i)T \)
19 \( 1 + (-0.233 + 0.972i)T \)
23 \( 1 + (0.382 + 0.923i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (-0.382 - 0.923i)T \)
41 \( 1 + (0.453 - 0.891i)T \)
43 \( 1 + (-0.522 - 0.852i)T \)
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 \( 1 + (-0.923 - 0.382i)T \)
73 \( 1 + (-0.852 - 0.522i)T \)
79 \( 1 + (0.156 + 0.987i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.0784 - 0.996i)T \)
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

−17.49750238772391123956172851990, −16.405999943852587958795391727318, −15.92182598571577755193223132874, −15.089946403994586019999348537566, −14.78009566593594455488439476873, −13.94630564061097620763287863241, −13.41660437925416144479867881694, −12.90474648479419467804962014125, −12.22095328607950413300700291140, −11.73974849926737998802039894190, −10.887267962818015716174316387203, −9.973444680426021716448598346155, −9.45404913740802388385259240189, −8.82930659832903970196601687303, −8.02226545561212963510858847590, −6.913451201075154418030601940357, −6.65615085059328802933685033644, −5.963704667103732356464832913206, −4.86633143774670363080419711578, −4.40725858260428965402447421318, −3.21232793353287881590841258351, −2.90689077605880478493398681798, −2.47060042370791251523640682905, −1.31554927770777336772244547689, −0.4971253436851374066279932910, 1.58592193983535277060526731156, 2.23355357124322593785995173460, 3.05529678796599875076416805635, 3.978143721350003118465010811805, 4.039468960650432066601860290265, 4.9389369394456622796078402792, 5.84865668227700684250519758267, 6.7121561981168811360430855357, 7.174662545977399273360487805246, 7.815370037478875475053019362559, 8.70250495158079512049510250751, 9.27821660022610998527320519713, 10.26812827472668863872178204564, 10.41928397002863127947771922106, 11.85257219924695512031944109717, 12.13405922634673694733046578700, 13.16598046380945744039660355235, 13.37069048353960616670463962572, 14.34286789392588025129522904807, 14.60680396974652442915899558945, 15.350598679400158545480738062011, 15.873257817203204831874951493984, 16.5988470158821133110914491628, 17.1360923411135816384017921895, 17.70592217348594032325363296864

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