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Properties

Label 1-5225-5225.4777-r0-0-0
Degree $1$
Conductor $5225$
Sign $-0.217 - 0.976i$
Analytic cond. $24.2647$
Root an. cond. $24.2647$
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  + (0.994 + 0.104i)2-s + (0.866 − 0.5i)3-s + (0.978 + 0.207i)4-s + (0.913 − 0.406i)6-s + (0.587 − 0.809i)7-s + (0.951 + 0.309i)8-s + (0.5 − 0.866i)9-s + (0.951 − 0.309i)12-s + (−0.866 − 0.5i)13-s + (0.669 − 0.743i)14-s + (0.913 + 0.406i)16-s + (−0.994 − 0.104i)17-s + (0.587 − 0.809i)18-s + (0.104 − 0.994i)21-s + (0.743 − 0.669i)23-s + (0.978 − 0.207i)24-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.866 − 0.5i)3-s + (0.978 + 0.207i)4-s + (0.913 − 0.406i)6-s + (0.587 − 0.809i)7-s + (0.951 + 0.309i)8-s + (0.5 − 0.866i)9-s + (0.951 − 0.309i)12-s + (−0.866 − 0.5i)13-s + (0.669 − 0.743i)14-s + (0.913 + 0.406i)16-s + (−0.994 − 0.104i)17-s + (0.587 − 0.809i)18-s + (0.104 − 0.994i)21-s + (0.743 − 0.669i)23-s + (0.978 − 0.207i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5225\)    =    \(5^{2} \cdot 11 \cdot 19\)
Sign: $-0.217 - 0.976i$
Analytic conductor: \(24.2647\)
Root analytic conductor: \(24.2647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5225} (4777, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5225,\ (0:\ ),\ -0.217 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.006150615 - 3.749159026i\)
\(L(\frac12)\) \(\approx\) \(3.006150615 - 3.749159026i\)
\(L(1)\) \(\approx\) \(2.416239471 - 0.8967679684i\)
\(L(1)\) \(\approx\) \(2.416239471 - 0.8967679684i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.994 + 0.104i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (-0.994 - 0.104i)T \)
23 \( 1 + (0.743 - 0.669i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.104 - 0.994i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.994 - 0.104i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.994 + 0.104i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (0.406 - 0.913i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (0.406 - 0.913i)T \)
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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

−18.364772521556931206622790644679, −17.315149309854798995499132859445, −16.63889198981712986477233325137, −15.83012404314152113350549707056, −15.26366217425284393367180844856, −14.86229375146187911409692107303, −14.26685448955485070190299696758, −13.608832800446185437909607512226, −12.942075806445616249788279886625, −12.27387051135223512390342480915, −11.45861153399323408073132091948, −10.98088748877557772330810114650, −10.144049062498532530864106824639, −9.326571978210178155096271962062, −8.78046702401649152867335388246, −7.90673411005832757895014506580, −7.21610636048888397331617711955, −6.52809940472005180295418110727, −5.43482092584297832644852062796, −4.90664711432425163230629875770, −4.42415684329785566155681203638, −3.41133773499475154024778473084, −2.91200681905593210714909028684, −1.91741027478939510950777865391, −1.71205808426698859558866341451, 0.62869396164813079214364770451, 1.813745572634318883514139279781, 2.24492426880182542292782167056, 3.1516459948265669215188507757, 3.84401187284995060439446744099, 4.545448823854862222018890415313, 5.19125877451390912312590998968, 6.19367834734222916266000272385, 7.042469594916005334684459041436, 7.33970843780251047962114447101, 8.05663091205923466149382107201, 8.79808638349087714361403756810, 9.733761119976294187998755255754, 10.57280182352935590413915727247, 11.15882116274601113244276205821, 12.011183791282846372036912914818, 12.6208801400755125015457564921, 13.41012334830871189886940231963, 13.59820053163447722885409263459, 14.48747318687448880647973251031, 15.035594822377869695969252501744, 15.304436753652274411069538800705, 16.46270422496372571251417249198, 17.01489761816042955510177646332, 17.7342922489124115001106331135

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