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Properties

Label 1-6025-6025.2583-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.877 - 0.480i$
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 + (−0.777 + 0.629i)3-s + (−0.207 − 0.978i)4-s + i·6-s + (−0.284 + 0.958i)7-s + (−0.891 − 0.453i)8-s + (0.207 − 0.978i)9-s + (0.958 + 0.284i)11-s + (0.777 + 0.629i)12-s + (−0.999 − 0.0261i)13-s + (0.566 + 0.824i)14-s + (−0.913 + 0.406i)16-s + (−0.972 − 0.233i)17-s + (−0.629 − 0.777i)18-s + (0.824 + 0.566i)19-s + ⋯
L(s)  = 1  + (0.629 − 0.777i)2-s + (−0.777 + 0.629i)3-s + (−0.207 − 0.978i)4-s + i·6-s + (−0.284 + 0.958i)7-s + (−0.891 − 0.453i)8-s + (0.207 − 0.978i)9-s + (0.958 + 0.284i)11-s + (0.777 + 0.629i)12-s + (−0.999 − 0.0261i)13-s + (0.566 + 0.824i)14-s + (−0.913 + 0.406i)16-s + (−0.972 − 0.233i)17-s + (−0.629 − 0.777i)18-s + (0.824 + 0.566i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.877 - 0.480i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (2583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.877 - 0.480i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.285530135 - 0.3290410706i\)
\(L(\frac12)\) \(\approx\) \(1.285530135 - 0.3290410706i\)
\(L(1)\) \(\approx\) \(0.9599401196 - 0.2189727614i\)
\(L(1)\) \(\approx\) \(0.9599401196 - 0.2189727614i\)

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 \( 1 + (-0.777 + 0.629i)T \)
7 \( 1 + (-0.284 + 0.958i)T \)
11 \( 1 + (0.958 + 0.284i)T \)
13 \( 1 + (-0.999 - 0.0261i)T \)
17 \( 1 + (-0.972 - 0.233i)T \)
19 \( 1 + (0.824 + 0.566i)T \)
23 \( 1 + (0.0784 - 0.996i)T \)
29 \( 1 + (-0.629 + 0.777i)T \)
31 \( 1 + (-0.725 - 0.688i)T \)
37 \( 1 + (0.333 + 0.942i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.233 - 0.972i)T \)
47 \( 1 + (0.156 - 0.987i)T \)
53 \( 1 + (0.358 + 0.933i)T \)
59 \( 1 + (-0.998 - 0.0523i)T \)
61 \( 1 + (-0.453 + 0.891i)T \)
67 \( 1 + (-0.933 - 0.358i)T \)
71 \( 1 + (0.878 - 0.477i)T \)
73 \( 1 + (0.522 + 0.852i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (0.878 - 0.477i)T \)
97 \( 1 + (-0.913 - 0.406i)T \)
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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

−17.57646382598981127852790438856, −17.142871060842521773617896822782, −16.40726413716385843099585329218, −16.05216194451642961031380441985, −15.11543631371869304633451975361, −14.350237466043292925130181660170, −13.80117898137782687008110121926, −13.19685056089246033066272134860, −12.69661929006337905016818669524, −11.92617334690794287693532038343, −11.29488667437757768028977460697, −10.822186356011367422961790241293, −9.50552282746249709469407250283, −9.23381342914489293013941566221, −7.84560377623718411081366266375, −7.5900057109087550844121208367, −6.83465837749188019722029325022, −6.41283569228217053318226482116, −5.65316358959376775797223106506, −4.8743402497074166635712235909, −4.28509496838589533808812396445, −3.53327689649170227270527014355, −2.591395224098815061525459679712, −1.5792976742584456838722005548, −0.5301586572757395662819874800, 0.54266867240262217048056125686, 1.68255197717293471944599788892, 2.42939059078746043894043612032, 3.256689881600124411402885085184, 4.034000536921721872866508907536, 4.652082563484055399087017654510, 5.32984144819347928767549873689, 5.90424785325478649491347706075, 6.597615133982155876446345642947, 7.28926731715260999355009897638, 8.81146757469644069414519051308, 9.22147546859296871765882557960, 9.752500206835856237103575673027, 10.48738572964073128727147841046, 11.1767024034060511418880055021, 11.83062716685144569331310177778, 12.296691278289860506999112845604, 12.66492483637335718400818132361, 13.71971894691085602659971930404, 14.46133370147024219470645296225, 15.189965237128034630742598412, 15.315504863844834309122589814701, 16.429908212337366672192337806194, 16.84134884057316881692337911952, 17.83198301133634950746071728864

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