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Properties

Label 1-6025-6025.2681-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.667 - 0.744i$
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  + (0.587 + 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.309 − 0.951i)6-s + (0.707 + 0.707i)7-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.987 − 0.156i)11-s + (0.587 − 0.809i)12-s + (−0.987 + 0.156i)13-s + (−0.156 + 0.987i)14-s + (−0.809 − 0.587i)16-s + (0.891 + 0.453i)17-s + i·18-s + (0.453 − 0.891i)19-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (−0.309 − 0.951i)6-s + (0.707 + 0.707i)7-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.987 − 0.156i)11-s + (0.587 − 0.809i)12-s + (−0.987 + 0.156i)13-s + (−0.156 + 0.987i)14-s + (−0.809 − 0.587i)16-s + (0.891 + 0.453i)17-s + i·18-s + (0.453 − 0.891i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.667 - 0.744i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (2681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.667 - 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3725775381 - 0.1662743747i\)
\(L(\frac12)\) \(\approx\) \(0.3725775381 - 0.1662743747i\)
\(L(1)\) \(\approx\) \(0.7613776520 + 0.4106228492i\)
\(L(1)\) \(\approx\) \(0.7613776520 + 0.4106228492i\)

Euler product

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

−17.75195233199472519548268058819, −17.27021446164553673094723760738, −16.5554241100261631330366708185, −15.63057197467162571222257958683, −15.29208504522876346452302226778, −14.23729396014729997230244236321, −13.953268648745993078275509641861, −13.07828554891505155015686014455, −12.22513103913730083857921920481, −11.953842762580500140450505640118, −11.33666341125637668526607848841, −10.328049636055530268372778608410, −10.16592274228585116343991420172, −9.75625487910679023552591167674, −8.41415020308214687414670048746, −7.6410223935266221496643613280, −6.95950610040637728849155350271, −5.94735009035984856973668513197, −5.3748295455939815834807297472, −4.822159141427151576906751970956, −4.26333777432785157477132318037, −3.44369774048525509629354248687, −2.58182697484374498870966644927, −1.632576955803727625131192549627, −0.883278726886008474559570152992, 0.1113758196388185583397128697, 1.45779131217628781039179230756, 2.47285847944829293765625633999, 3.08349022438310496971683211158, 4.40514239324243151873862142165, 4.87768913485267489151664181949, 5.36402272130567131085086566044, 6.03795314814402742960811923608, 6.677955833541952160205178164749, 7.63745653506556369655264438241, 7.858067509497663805846988214762, 8.68405514897558673409961194753, 9.67477616933771216091072844432, 10.40763824863523294171837848177, 11.24612394128186658656491265204, 12.00840863026717424279804746273, 12.27321440418346004935501500072, 12.95060577814390397332197741227, 13.78021314097837251507625886244, 14.3279789035569602003017562957, 15.122323912435330558810717171022, 15.74316339985933988974507062670, 16.20111946272990431463231777251, 16.95656361882953918721001285482, 17.6558863063048967612598329389

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