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Properties

Label 1-6025-6025.13-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.924 + 0.380i$
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  + (0.544 + 0.838i)2-s + (−0.258 + 0.965i)3-s + (−0.406 + 0.913i)4-s + (−0.951 + 0.309i)6-s + (0.942 − 0.333i)7-s + (−0.987 + 0.156i)8-s + (−0.866 − 0.5i)9-s + (0.824 + 0.566i)11-s + (−0.777 − 0.629i)12-s + (−0.0261 + 0.999i)13-s + (0.793 + 0.608i)14-s + (−0.669 − 0.743i)16-s + (0.852 + 0.522i)17-s + (−0.0523 − 0.998i)18-s + (−0.0261 − 0.999i)19-s + ⋯
L(s)  = 1  + (0.544 + 0.838i)2-s + (−0.258 + 0.965i)3-s + (−0.406 + 0.913i)4-s + (−0.951 + 0.309i)6-s + (0.942 − 0.333i)7-s + (−0.987 + 0.156i)8-s + (−0.866 − 0.5i)9-s + (0.824 + 0.566i)11-s + (−0.777 − 0.629i)12-s + (−0.0261 + 0.999i)13-s + (0.793 + 0.608i)14-s + (−0.669 − 0.743i)16-s + (0.852 + 0.522i)17-s + (−0.0523 − 0.998i)18-s + (−0.0261 − 0.999i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.924 + 0.380i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.924 + 0.380i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5043272625 + 2.551609258i\)
\(L(\frac12)\) \(\approx\) \(0.5043272625 + 2.551609258i\)
\(L(1)\) \(\approx\) \(0.9404652592 + 1.105641512i\)
\(L(1)\) \(\approx\) \(0.9404652592 + 1.105641512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.544 + 0.838i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 + (0.942 - 0.333i)T \)
11 \( 1 + (0.824 + 0.566i)T \)
13 \( 1 + (-0.0261 + 0.999i)T \)
17 \( 1 + (0.852 + 0.522i)T \)
19 \( 1 + (-0.0261 - 0.999i)T \)
23 \( 1 + (-0.760 - 0.649i)T \)
29 \( 1 + (0.933 + 0.358i)T \)
31 \( 1 + (0.477 + 0.878i)T \)
37 \( 1 + (0.566 - 0.824i)T \)
41 \( 1 + (0.987 - 0.156i)T \)
43 \( 1 + (0.649 + 0.760i)T \)
47 \( 1 + (-0.987 + 0.156i)T \)
53 \( 1 + (0.777 - 0.629i)T \)
59 \( 1 + (-0.838 + 0.544i)T \)
61 \( 1 + (-0.891 + 0.453i)T \)
67 \( 1 + (0.629 - 0.777i)T \)
71 \( 1 + (0.688 - 0.725i)T \)
73 \( 1 + (-0.852 + 0.522i)T \)
79 \( 1 + (0.453 + 0.891i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.983 + 0.182i)T \)
97 \( 1 + (0.104 - 0.994i)T \)
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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

−17.56406477694550223960082089048, −17.06481430240235588523491500921, −16.09494873681250076694582387046, −15.16939420270411969520555886719, −14.53293313349681752780180279347, −13.93471311326423019875841829109, −13.60251078950091459729926135431, −12.55985703734384132082327579196, −12.18029086359622098524264148818, −11.58668215338868773115636796195, −11.160325393641421832510022858290, −10.273499500159036643161682935448, −9.62327061039212272999058130336, −8.62500573419124703226652278839, −8.05111660325347592323774168457, −7.46896371516611170889054447414, −6.167810202952184613866093447394, −5.946133477671499546498556425294, −5.24290769278585751971742330180, −4.43497914562809871361598668445, −3.5074657782692047378681132076, −2.79020083845613334686496489527, −2.03054209587556777630651640336, −1.23181246993607685867593474512, −0.735738479932518392088272037812, 0.918197720933120971946673514317, 2.11943573708859944975247520622, 3.1103378989044017947059379435, 3.98048924159525980640436929975, 4.53025109606895181247737900995, 4.79308781583565585282425066447, 5.80420133075016944132405506556, 6.42621701374879913011731954832, 7.1084741577262183943566009659, 7.93884413354509703579274086439, 8.65986121101296915583816914061, 9.20671349213214458116195841242, 9.93943245671727682352487511478, 10.81205642788389384039120318021, 11.48877070427063489334560340392, 12.08459972251250667656774720534, 12.63554995897845531394849749715, 13.88954235148448448918639354640, 14.24888479518431102842485349982, 14.66005637467105106556887785782, 15.31541720733327095742756616766, 16.10463136982130876229810889412, 16.58625345093404867063283862028, 17.137041761795092057682769181885, 17.80406026991863550837766473744

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