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Properties

Label 1-6025-6025.278-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.539 + 0.841i$
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  + (0.629 + 0.777i)2-s + (0.358 − 0.933i)3-s + (−0.207 + 0.978i)4-s + (0.951 − 0.309i)6-s + (0.942 + 0.333i)7-s + (−0.891 + 0.453i)8-s + (−0.743 − 0.669i)9-s + (0.284 + 0.958i)11-s + (0.838 + 0.544i)12-s + (0.942 − 0.333i)13-s + (0.333 + 0.942i)14-s + (−0.913 − 0.406i)16-s + (−0.382 + 0.923i)17-s + (0.0523 − 0.998i)18-s + (0.566 + 0.824i)19-s + ⋯
L(s)  = 1  + (0.629 + 0.777i)2-s + (0.358 − 0.933i)3-s + (−0.207 + 0.978i)4-s + (0.951 − 0.309i)6-s + (0.942 + 0.333i)7-s + (−0.891 + 0.453i)8-s + (−0.743 − 0.669i)9-s + (0.284 + 0.958i)11-s + (0.838 + 0.544i)12-s + (0.942 − 0.333i)13-s + (0.333 + 0.942i)14-s + (−0.913 − 0.406i)16-s + (−0.382 + 0.923i)17-s + (0.0523 − 0.998i)18-s + (0.566 + 0.824i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.539 + 0.841i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.539 + 0.841i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.358493854 + 2.485388579i\)
\(L(\frac12)\) \(\approx\) \(1.358493854 + 2.485388579i\)
\(L(1)\) \(\approx\) \(1.529770993 + 0.7179061690i\)
\(L(1)\) \(\approx\) \(1.529770993 + 0.7179061690i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.629 + 0.777i)T \)
3 \( 1 + (0.358 - 0.933i)T \)
7 \( 1 + (0.942 + 0.333i)T \)
11 \( 1 + (0.284 + 0.958i)T \)
13 \( 1 + (0.942 - 0.333i)T \)
17 \( 1 + (-0.382 + 0.923i)T \)
19 \( 1 + (0.566 + 0.824i)T \)
23 \( 1 + (0.382 + 0.923i)T \)
29 \( 1 + (-0.965 - 0.258i)T \)
31 \( 1 + (-0.130 - 0.991i)T \)
37 \( 1 + (-0.608 + 0.793i)T \)
41 \( 1 + (0.156 + 0.987i)T \)
43 \( 1 + (0.649 - 0.760i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (0.258 - 0.965i)T \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 + (0.965 + 0.258i)T \)
71 \( 1 + (0.130 + 0.991i)T \)
73 \( 1 + (-0.760 + 0.649i)T \)
79 \( 1 + (-0.891 + 0.453i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (0.477 - 0.878i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−17.68369677985466183990303117830, −16.5475053972072531598403682593, −16.09312316893111925150590979312, −15.44553597541970627555638334012, −14.61346498701208869499311771452, −14.14937129984320804271612689252, −13.73126175600053013634620154121, −13.08639300500510002132648110967, −12.00026350140306046045770284721, −11.2851047106232663870947045467, −10.96621954181310649585709231422, −10.55589374159376162322313404410, −9.4409337681805005262956831137, −8.96744716470596363705734250352, −8.50492547631071316400813279795, −7.40915421063372458471807551171, −6.50280010760516334725465704056, −5.596738174664293856663163158408, −5.01460445560871270395085058854, −4.444735733456234653784688760902, −3.6954376738620171986068059127, −3.148549423855279933783971820102, −2.337712442005424343849972573806, −1.45058276430999357385250794817, −0.51170050236808978959095730113, 1.299361757429241841909976476382, 1.849403253744584201449926785, 2.72732482551770304156888027479, 3.71282536832213370240980724187, 4.14970554226886066620992358976, 5.3182717902590249233212307179, 5.72268418920303693602387953219, 6.4780145312482055695928340178, 7.21741490596663279936409281930, 7.846025518966794984415894387521, 8.26454146847186499537186872013, 8.96578671324040273962410685636, 9.7090600952730255576524571772, 10.97464916602068352335407389338, 11.6354912277833293783085911603, 12.11917455387286278029460453134, 12.92329897451252993041066857680, 13.32965209420952854174407508133, 14.02798413962737312003804501779, 14.75166624140798817054741978017, 15.099788261193963975008887107817, 15.68006564814422745407471496224, 16.7717987149891174699637576128, 17.4137040314419996085642739652, 17.704666916589698355031928197543

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