↑

Properties

Label 1-6025-6025.1353-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.213 - 0.976i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.987 − 0.156i)2-s + (0.987 − 0.156i)3-s + (0.951 − 0.309i)4-s + (0.951 − 0.309i)6-s + (−0.649 − 0.760i)7-s + (0.891 − 0.453i)8-s + (0.951 − 0.309i)9-s + (0.233 − 0.972i)11-s + (0.891 − 0.453i)12-s + (−0.649 + 0.760i)13-s + (−0.760 − 0.649i)14-s + (0.809 − 0.587i)16-s + (0.923 + 0.382i)17-s + (0.891 − 0.453i)18-s + (−0.0784 − 0.996i)19-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s + (0.987 − 0.156i)3-s + (0.951 − 0.309i)4-s + (0.951 − 0.309i)6-s + (−0.649 − 0.760i)7-s + (0.891 − 0.453i)8-s + (0.951 − 0.309i)9-s + (0.233 − 0.972i)11-s + (0.891 − 0.453i)12-s + (−0.649 + 0.760i)13-s + (−0.760 − 0.649i)14-s + (0.809 − 0.587i)16-s + (0.923 + 0.382i)17-s + (0.891 − 0.453i)18-s + (−0.0784 − 0.996i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.213 - 0.976i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.213 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.182596335 - 3.954859037i\)
\(L(\frac12)\) \(\approx\) \(3.182596335 - 3.954859037i\)
\(L(1)\) \(\approx\) \(2.393053736 - 1.030991882i\)
\(L(1)\) \(\approx\) \(2.393053736 - 1.030991882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.987 - 0.156i)T \)
3 \( 1 + (0.987 - 0.156i)T \)
7 \( 1 + (-0.649 - 0.760i)T \)
11 \( 1 + (0.233 - 0.972i)T \)
13 \( 1 + (-0.649 + 0.760i)T \)
17 \( 1 + (0.923 + 0.382i)T \)
19 \( 1 + (-0.0784 - 0.996i)T \)
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 \( 1 + (-0.156 - 0.987i)T \)
43 \( 1 + (-0.760 - 0.649i)T \)
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 \( 1 + (-0.649 - 0.760i)T \)
79 \( 1 + (0.891 - 0.453i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.852 - 0.522i)T \)
97 \( 1 + T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.01433402633884297819127896159, −16.94594429037410097509065833649, −16.32038251104487866886961137224, −15.74858271004202176411952497607, −15.00144966064686916579508009966, −14.730412736405277994839197028922, −14.144591693537609548232267975979, −13.20143956193268997709729457584, −12.8034001775389060699525909170, −12.1178329184529309635601931141, −11.74526977221568869056082827428, −10.30060722119457969862134787618, −10.026602256030334076207976890762, −9.38620734981973877632262325595, −8.23974847348917486234566957326, −7.879975072612756567144762198924, −7.125260529778849504586430981267, −6.36916128327624835122137050430, −5.61390322072040427500400744003, −4.88692827519041638300611479222, −4.15021631360087174708565696895, −3.408287785489249876871264876114, −2.7987386790145055333282255831, −2.20289284613815648504214717684, −1.39976273693063618136991900143, 0.68244977765017541588572393734, 1.62620892242018375022130952601, 2.40385325695590102669398608498, 3.17234410462649882447510602721, 3.75174235530255632898676225143, 4.22709024082570778903850937743, 5.16201864394667738035858457913, 6.14220318224304257042975595202, 6.62747511078256266601062939106, 7.49761998586611224088378234947, 7.80058128472190899655406094193, 8.98952194336935795973961488409, 9.567297787200873344095250935646, 10.24772581356573492745617277176, 10.973083957450004889351174160773, 11.74957148517135643115268242182, 12.473323356146459755084732735841, 13.094678034418524864838069619896, 13.660532711206857558795827799, 14.19968127057802587077874986424, 14.56626827999606761036971590967, 15.52385007608054815363881155631, 15.97068502398498898303752610333, 16.75746859023049124423639464496, 17.21066513636037375090330955542

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