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Properties

Label 1-6025-6025.2578-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.975 - 0.217i$
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  + (−0.987 + 0.156i)2-s + (−0.987 + 0.156i)3-s + (0.951 − 0.309i)4-s + (0.951 − 0.309i)6-s + (0.760 − 0.649i)7-s + (−0.891 + 0.453i)8-s + (0.951 − 0.309i)9-s + (0.972 + 0.233i)11-s + (−0.891 + 0.453i)12-s + (0.760 + 0.649i)13-s + (−0.649 + 0.760i)14-s + (0.809 − 0.587i)16-s + (0.382 − 0.923i)17-s + (−0.891 + 0.453i)18-s + (0.996 − 0.0784i)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 + (0.760 − 0.649i)7-s + (−0.891 + 0.453i)8-s + (0.951 − 0.309i)9-s + (0.972 + 0.233i)11-s + (−0.891 + 0.453i)12-s + (0.760 + 0.649i)13-s + (−0.649 + 0.760i)14-s + (0.809 − 0.587i)16-s + (0.382 − 0.923i)17-s + (−0.891 + 0.453i)18-s + (0.996 − 0.0784i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.975 - 0.217i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (2578, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.975 - 0.217i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.148693891 - 0.1266554989i\)
\(L(\frac12)\) \(\approx\) \(1.148693891 - 0.1266554989i\)
\(L(1)\) \(\approx\) \(0.7069683027 + 0.003309404718i\)
\(L(1)\) \(\approx\) \(0.7069683027 + 0.003309404718i\)

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 \( 1 + (0.760 - 0.649i)T \)
11 \( 1 + (0.972 + 0.233i)T \)
13 \( 1 + (0.760 + 0.649i)T \)
17 \( 1 + (0.382 - 0.923i)T \)
19 \( 1 + (0.996 - 0.0784i)T \)
23 \( 1 + (-0.382 - 0.923i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (0.156 + 0.987i)T \)
43 \( 1 + (-0.649 + 0.760i)T \)
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 \( 1 + (0.923 + 0.382i)T \)
73 \( 1 + (0.760 - 0.649i)T \)
79 \( 1 + (-0.891 + 0.453i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.522 - 0.852i)T \)
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

−17.844394341253309513058266385957, −17.21669876674760978814678599609, −16.62516204970768394097580204612, −15.86202193263557761022063672859, −15.49043607258713213265755661942, −14.59524807799086356414044322062, −13.83312315112623438389487880900, −12.67410386759480953041858177830, −12.31743222577627684908983767762, −11.59816397445268411695812308097, −11.13217840278987856782602177027, −10.57742031051081418621599321635, −9.81301221562283561510046498242, −9.04445447111583091309842054138, −8.43483577213456698499622511338, −7.67477776965431151438388280478, −7.114795612870932495564664924566, −6.11987347041882715168763398924, −5.79505040101581474870916372053, −5.05611468062793375392002736351, −3.82813947946369524567911337345, −3.29599915239262833079399673559, −1.94363223033232269001492378227, −1.48099069953309918451135891094, −0.78478573774333314674022653233, 0.72343435758209296919454153873, 1.22409233626575478698194199116, 1.95601287903870392162590888806, 3.1888414344383395415990335633, 4.15409855906272341127933182541, 4.7724839103383327816653202570, 5.63907105072247242906918757433, 6.41032996745049563593606880455, 6.90841713894152645211650249276, 7.53849866563599447082067566977, 8.32982366347083848249267498350, 9.105022183650007406620007303437, 9.93145435749839829188465460612, 10.16883101367864888567363947434, 11.278654079854094355625783891745, 11.56798320292643342083323788567, 11.85553962205443513327362626195, 12.997413303667895343740492973114, 13.92453810789930523595413002656, 14.53247360751097810670760197917, 15.24292402228269134083768059321, 16.10416559806304446159059373390, 16.65128733724513250843708298731, 16.82441031959468058355807795781, 17.78115892020765904023829396706

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