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Properties

Label 1-35e2-1225.414-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.999 + 0.0282i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (0.550 + 0.834i)2-s + (−0.473 − 0.880i)3-s + (−0.393 + 0.919i)4-s + (0.473 − 0.880i)6-s + (−0.983 + 0.178i)8-s + (−0.550 + 0.834i)9-s + (−0.550 − 0.834i)11-s + (0.995 − 0.0896i)12-s + (−0.936 − 0.351i)13-s + (−0.691 − 0.722i)16-s + (0.393 + 0.919i)17-s − 18-s + (−0.809 + 0.587i)19-s + (0.393 − 0.919i)22-s + (0.995 + 0.0896i)23-s + (0.623 + 0.781i)24-s + ⋯
L(s)  = 1  + (0.550 + 0.834i)2-s + (−0.473 − 0.880i)3-s + (−0.393 + 0.919i)4-s + (0.473 − 0.880i)6-s + (−0.983 + 0.178i)8-s + (−0.550 + 0.834i)9-s + (−0.550 − 0.834i)11-s + (0.995 − 0.0896i)12-s + (−0.936 − 0.351i)13-s + (−0.691 − 0.722i)16-s + (0.393 + 0.919i)17-s − 18-s + (−0.809 + 0.587i)19-s + (0.393 − 0.919i)22-s + (0.995 + 0.0896i)23-s + (0.623 + 0.781i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0282i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (414, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.999 + 0.0282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217862173 + 0.01717921907i\)
\(L(\frac12)\) \(\approx\) \(1.217862173 + 0.01717921907i\)
\(L(1)\) \(\approx\) \(0.9931323235 + 0.1768468756i\)
\(L(1)\) \(\approx\) \(0.9931323235 + 0.1768468756i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.550 + 0.834i)T \)
3 \( 1 + (-0.473 - 0.880i)T \)
11 \( 1 + (-0.550 - 0.834i)T \)
13 \( 1 + (-0.936 - 0.351i)T \)
17 \( 1 + (0.393 + 0.919i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.995 + 0.0896i)T \)
29 \( 1 + (0.753 - 0.657i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (0.995 - 0.0896i)T \)
41 \( 1 + (0.134 - 0.990i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.963 - 0.266i)T \)
53 \( 1 + (0.393 - 0.919i)T \)
59 \( 1 + (-0.691 - 0.722i)T \)
61 \( 1 + (0.858 + 0.512i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (-0.393 + 0.919i)T \)
73 \( 1 + (-0.936 + 0.351i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.963 + 0.266i)T \)
89 \( 1 + (0.936 - 0.351i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−21.264775470161431912895249436935, −20.455193196046115581228865688468, −19.94440852720202215736151912672, −18.95845014348349950828935777803, −18.10662235800383667372880597936, −17.368867056840108609970843660885, −16.4811811007506397653811596607, −15.551801651586958197303402415, −14.84688045479303891629994446869, −14.3609135173178385704939873504, −13.15617195613046469292632468478, −12.46559443428164225388717701848, −11.730245606414623023405609035611, −10.95006907493333258883148117944, −10.29854452540616677424017233853, −9.4777194431678379131422257223, −9.017586652264726541524912717824, −7.46623306461667019713793684216, −6.47943069359505799665104795549, −5.43300293129697114509352542257, −4.72762052663728210902690903400, −4.2843828559690768490499133978, −2.97575568276971898401281183107, −2.39622635940302942640427459906, −0.852320172022364190528846153741, 0.5739795633828921454669054631, 2.17273613762400985473845063936, 3.07584042938183015724552088756, 4.2206941267580146412643646969, 5.35149290071124763407518921023, 5.76975898712691848375142056270, 6.675848428852795215112916911996, 7.48510624207278870971821872052, 8.14453418914075293242925909984, 8.85892313739113218863244027469, 10.28074180190608482312861925234, 11.10803921874906868207421804464, 12.14242482914864747384055868031, 12.70639343955885193994452998308, 13.277000213059140735215295762597, 14.21480795639756118525662561269, 14.80769199968650860043276490240, 15.799976543913355019365297453030, 16.647528758840791980431711662400, 17.20870257818470936521382042736, 17.79655968967120108682081664856, 18.882264069724844218359722596723, 19.21870067075691637695668902039, 20.47973929983773389189383363185, 21.52874926184693765598340209839

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