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Properties

Label 1-6025-6025.814-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.0459 - 0.998i$
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  + (−0.309 + 0.951i)2-s − 3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)6-s + (−0.309 − 0.951i)7-s + (0.809 − 0.587i)8-s + 9-s + (0.309 − 0.951i)11-s + (0.809 + 0.587i)12-s + (0.809 + 0.587i)13-s + 14-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)17-s + (−0.309 + 0.951i)18-s + (−0.809 + 0.587i)19-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s − 3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)6-s + (−0.309 − 0.951i)7-s + (0.809 − 0.587i)8-s + 9-s + (0.309 − 0.951i)11-s + (0.809 + 0.587i)12-s + (0.809 + 0.587i)13-s + 14-s + (0.309 + 0.951i)16-s + (0.809 + 0.587i)17-s + (−0.309 + 0.951i)18-s + (−0.809 + 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0459 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0459 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.0459 - 0.998i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (814, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.0459 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2988288238 - 0.2853856905i\)
\(L(\frac12)\) \(\approx\) \(0.2988288238 - 0.2853856905i\)
\(L(1)\) \(\approx\) \(0.5650279067 + 0.1276036074i\)
\(L(1)\) \(\approx\) \(0.5650279067 + 0.1276036074i\)

Euler product

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

−17.97033250518386152477406941609, −17.34925660421683190046871206529, −16.85143060481573800429544387684, −15.898884150174729269892101745941, −15.47284106247343116649773125469, −14.62794072175225673499532688562, −13.58507463159690847236659144429, −12.99411637491252486879718245117, −12.364134242241672309150357629, −11.96773393287357805245643456537, −11.33495813870109272653389604044, −10.59926393603617953068788385463, −10.08560074619709680385333203215, −9.25085829177409540945704159924, −8.92159395892220494095041857956, −7.77870860591176050551568342525, −7.221086769243147968789803924153, −6.276573237067023088721157310277, −5.389557098635450585199303818889, −5.1268859747889827036501702514, −3.95559703953343532251459460443, −3.5514868918592694141400192653, −2.367073588757958925992520723942, −1.814148110547743680636493289241, −0.88298343163534806201472048922, 0.1843504279694236809358096154, 1.11151771121989025495747620369, 1.7060403879952604772351190477, 3.56154066341387361660718171620, 3.93498373311104921967200265113, 4.67983460875337699134212745726, 5.67309297442136201640761965495, 6.06690422229405118158196906557, 6.715642311379650868223711873603, 7.219314993021145788613343793647, 8.23174676566555953018794182169, 8.61112594832026836346569278495, 9.74484123682110135016587011426, 10.16015382149366974659613115283, 10.8845599298003384639250962428, 11.36155125938165962450294919327, 12.42574625582472260078578800234, 13.11062396649518450313971655995, 13.59302228833546028872845302862, 14.44260203712569847241427829797, 14.92366777799082377255334449487, 15.98354859011206205869575637502, 16.43631867444046507267208609894, 16.84607935665199103818929391451, 17.13166265194242535897896024843

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