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Properties

Label 1-6025-6025.481-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.929 - 0.368i$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s − 7-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)12-s + (0.809 + 0.587i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + 18-s + (−0.309 − 0.951i)19-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s − 7-s + (0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)12-s + (0.809 + 0.587i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + 18-s + (−0.309 − 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.929 - 0.368i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1729916200 - 0.9068538304i\)
\(L(\frac12)\) \(\approx\) \(0.1729916200 - 0.9068538304i\)
\(L(1)\) \(\approx\) \(0.6854330110 - 0.2660508149i\)
\(L(1)\) \(\approx\) \(0.6854330110 - 0.2660508149i\)

Euler product

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

−18.020822699238789656568250523467, −17.22598233439434951688828740458, −16.68376882174023186924681142689, −16.25374392992190829552077567058, −15.38509551444283195748923422661, −15.063588142899794099939470862791, −14.04243047405278025095307287972, −13.26881417893355147044906297670, −12.575518183926854126891382470674, −12.08488799332882785814602315979, −10.9663160589604695408162118998, −10.70316056783840513197743170703, −10.00539114172576684271814419935, −9.36635092003128922242620783852, −8.89659881789626476942814596128, −8.26154475778925645273168333069, −7.46315412488917029735835815434, −6.52485357423063366878853742277, −5.99376711999130247755777513875, −4.84230303096747076358420587646, −3.972113783855841529673310161329, −3.445307849803740661750955589517, −3.00608192317432153315290446515, −1.89362136256525097108563721721, −1.16410703559884793234081857694, 0.37073439871617247603251349364, 0.89941149314726457159326898841, 1.94902973974419704514840592010, 2.66001016788646145199211208996, 3.46134943797538371115090330145, 4.49631028792718874547818046880, 5.57842119219754640527693867314, 6.33070149532504043551251836812, 6.69749490823199689358688782650, 7.13412373137499552650969033725, 8.136653387856349449322480836251, 8.779946967358010937540848706530, 9.171252885841084892839631394872, 9.79760600774606186719640738077, 10.83509363164063536701469109034, 11.5421056932844187913497404504, 11.90894686359083235197202296404, 13.23819689203582343950725565535, 13.41240532110709772046080410386, 14.08365455023131029950515439517, 14.94931607026055678384423542868, 15.4651556392716093752441702125, 16.33407545833036870506126805086, 16.81480761429097740168560698391, 17.342860681155661952382610808035

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