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Properties

Label 1-6025-6025.3763-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.787 + 0.616i$
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 + (0.453 + 0.891i)3-s + (0.587 + 0.809i)4-s − i·6-s + (−0.649 − 0.760i)7-s + (−0.156 − 0.987i)8-s + (−0.587 + 0.809i)9-s + (−0.760 + 0.649i)11-s + (−0.453 + 0.891i)12-s + (0.0784 − 0.996i)13-s + (0.233 + 0.972i)14-s + (−0.309 + 0.951i)16-s + (0.649 − 0.760i)17-s + (0.891 − 0.453i)18-s + (−0.972 − 0.233i)19-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)2-s + (0.453 + 0.891i)3-s + (0.587 + 0.809i)4-s − i·6-s + (−0.649 − 0.760i)7-s + (−0.156 − 0.987i)8-s + (−0.587 + 0.809i)9-s + (−0.760 + 0.649i)11-s + (−0.453 + 0.891i)12-s + (0.0784 − 0.996i)13-s + (0.233 + 0.972i)14-s + (−0.309 + 0.951i)16-s + (0.649 − 0.760i)17-s + (0.891 − 0.453i)18-s + (−0.972 − 0.233i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.787 + 0.616i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (3763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.787 + 0.616i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6767994857 + 0.2333671561i\)
\(L(\frac12)\) \(\approx\) \(0.6767994857 + 0.2333671561i\)
\(L(1)\) \(\approx\) \(0.6454717054 + 0.03335145357i\)
\(L(1)\) \(\approx\) \(0.6454717054 + 0.03335145357i\)

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 \( 1 + (0.453 + 0.891i)T \)
7 \( 1 + (-0.649 - 0.760i)T \)
11 \( 1 + (-0.760 + 0.649i)T \)
13 \( 1 + (0.0784 - 0.996i)T \)
17 \( 1 + (0.649 - 0.760i)T \)
19 \( 1 + (-0.972 - 0.233i)T \)
23 \( 1 + (0.972 + 0.233i)T \)
29 \( 1 + (-0.891 - 0.453i)T \)
31 \( 1 + (-0.760 - 0.649i)T \)
37 \( 1 + (-0.522 + 0.852i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (-0.760 - 0.649i)T \)
47 \( 1 + (-0.453 + 0.891i)T \)
53 \( 1 + (-0.891 - 0.453i)T \)
59 \( 1 + (0.987 + 0.156i)T \)
61 \( 1 + (-0.987 + 0.156i)T \)
67 \( 1 + (-0.453 - 0.891i)T \)
71 \( 1 + (-0.996 - 0.0784i)T \)
73 \( 1 + (0.0784 + 0.996i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (-0.996 - 0.0784i)T \)
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

−17.80841557563287713333192199926, −16.91497258671857071711831415068, −16.47401008423007228085886199203, −15.80873305976236128211635885184, −14.855525923744105554161775092525, −14.67415874363866877884446876183, −13.78351240512618285494912062495, −12.92499646786156067030479300556, −12.52505815433940914463032472981, −11.64800329198897719313336929361, −10.95370592478065849224264756088, −10.26813688814759420842681646417, −9.32269934793828173418235996358, −8.77031550538146981911257630023, −8.49689359843618841250133099628, −7.51879988880231509995360323353, −7.04406841372991803983326317208, −6.16856894837073477946443112403, −5.891380053253915543068103720234, −4.99796094390043463509691436198, −3.58441157037044988079466126602, −2.95627679210555229441643322761, −2.0183685493181073291728104611, −1.629278220916768900326508226532, −0.37942821488438015366055296127, 0.54098538856672089180339773888, 1.728184145194748757167289887295, 2.706039801027252944894930906278, 3.124683170342669839259129126162, 3.8438303866502433748990278107, 4.65872664537323497550913435268, 5.459103772576918220295446796212, 6.47591553507190011768168944727, 7.42541800239502020784648466148, 7.7520388206528827074241733094, 8.52614947239981832803272222680, 9.41961863467363614460249805927, 9.75346874386315944364587813781, 10.39863516991390199673825135418, 10.882162813782938411490831500811, 11.51208485200055280538356307776, 12.684281624856456886962691598085, 13.02032941073313890355719425729, 13.69051086857553759840676101226, 14.82356014925160732620903902631, 15.29493616679355002366484990873, 15.87660431762024639314050279723, 16.6529806318471801031351015473, 16.98164647114222849201057124387, 17.72509330430987642095031562394

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