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Properties

Label 1-6025-6025.4777-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.158 + 0.987i$
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 + (−0.891 + 0.453i)3-s + (0.587 − 0.809i)4-s + (0.587 − 0.809i)6-s + (−0.522 − 0.852i)7-s + (−0.156 + 0.987i)8-s + (0.587 − 0.809i)9-s + (0.760 + 0.649i)11-s + (−0.156 + 0.987i)12-s + (−0.522 + 0.852i)13-s + (0.852 + 0.522i)14-s + (−0.309 − 0.951i)16-s + (−0.923 + 0.382i)17-s + (−0.156 + 0.987i)18-s + (0.972 − 0.233i)19-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)2-s + (−0.891 + 0.453i)3-s + (0.587 − 0.809i)4-s + (0.587 − 0.809i)6-s + (−0.522 − 0.852i)7-s + (−0.156 + 0.987i)8-s + (0.587 − 0.809i)9-s + (0.760 + 0.649i)11-s + (−0.156 + 0.987i)12-s + (−0.522 + 0.852i)13-s + (0.852 + 0.522i)14-s + (−0.309 − 0.951i)16-s + (−0.923 + 0.382i)17-s + (−0.156 + 0.987i)18-s + (0.972 − 0.233i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.158 + 0.987i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4777, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.158 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4442853573 + 0.3785071859i\)
\(L(\frac12)\) \(\approx\) \(0.4442853573 + 0.3785071859i\)
\(L(1)\) \(\approx\) \(0.4919088869 + 0.1413760280i\)
\(L(1)\) \(\approx\) \(0.4919088869 + 0.1413760280i\)

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 \( 1 + (-0.891 + 0.453i)T \)
7 \( 1 + (-0.522 - 0.852i)T \)
11 \( 1 + (0.760 + 0.649i)T \)
13 \( 1 + (-0.522 + 0.852i)T \)
17 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (0.972 - 0.233i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (-0.453 - 0.891i)T \)
43 \( 1 + (0.852 + 0.522i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (-0.522 - 0.852i)T \)
79 \( 1 + (-0.156 + 0.987i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.996 - 0.0784i)T \)
97 \( 1 + T \)
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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

−17.46066054031929861255876959332, −17.23125548385738470801260011337, −16.30575078782484969721569561592, −15.92541279789643983407344745408, −15.27333799906921666311502412811, −14.222828335839062913852832258271, −13.2726698183039812681969465530, −12.68977910880421241065820746948, −12.17394584523916792423663512637, −11.62056783028921534760537781766, −10.911066941203736688147460055562, −10.46690696772987126199190518460, −9.4192536288247439165969215547, −9.11195458703431419936697920487, −8.25752249734939561700512901897, −7.45693128899285853256796649808, −6.8113970838777941257685729623, −6.26418365829465199811367736400, −5.43498162904169322408596195014, −4.757083771879139538129159364519, −3.4550582555005765651259786728, −2.9680885886630159317335068159, −2.03920434095124592246445083989, −1.24416343235832013318784271339, −0.38726120283317647491580426673, 0.62480849377435813665871200897, 1.47368060440589167398196072133, 2.32874251548832312749237256283, 3.66020996761212794975432512365, 4.2538334287338369447029271655, 5.04409349830556427234090373126, 5.81446147507200019384294791691, 6.58795571360746613439514090774, 7.14613946047979900490154188464, 7.39878460496193110567284502867, 8.77400954758607854599076043827, 9.43895977722838268406295788697, 9.69025577620283466941401486619, 10.48910095723936752881948581123, 11.18305794958903564546084277460, 11.61022650022786195969818742183, 12.399525192980705538994591469267, 13.298359993953452402004780134852, 14.09040625038768491846901155389, 14.86856039509453449834896720290, 15.44708866045852110873409432790, 16.06124000418673451969858834248, 16.72336858008224388244076341368, 17.30418844917112765056464979991, 17.41451792497217706154053468655

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