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Properties

Label 1-725-725.272-r0-0-0
Degree $1$
Conductor $725$
Sign $0.856 - 0.516i$
Analytic cond. $3.36688$
Root an. cond. $3.36688$
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  + (−0.0448 − 0.998i)2-s + (−0.858 + 0.512i)3-s + (−0.995 + 0.0896i)4-s + (0.550 + 0.834i)6-s + (0.974 − 0.222i)7-s + (0.134 + 0.990i)8-s + (0.473 − 0.880i)9-s + (0.880 − 0.473i)11-s + (0.809 − 0.587i)12-s + (0.178 + 0.983i)13-s + (−0.266 − 0.963i)14-s + (0.983 − 0.178i)16-s + (0.309 − 0.951i)17-s + (−0.900 − 0.433i)18-s + (−0.512 + 0.858i)19-s + ⋯
L(s)  = 1  + (−0.0448 − 0.998i)2-s + (−0.858 + 0.512i)3-s + (−0.995 + 0.0896i)4-s + (0.550 + 0.834i)6-s + (0.974 − 0.222i)7-s + (0.134 + 0.990i)8-s + (0.473 − 0.880i)9-s + (0.880 − 0.473i)11-s + (0.809 − 0.587i)12-s + (0.178 + 0.983i)13-s + (−0.266 − 0.963i)14-s + (0.983 − 0.178i)16-s + (0.309 − 0.951i)17-s + (−0.900 − 0.433i)18-s + (−0.512 + 0.858i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.856 - 0.516i$
Analytic conductor: \(3.36688\)
Root analytic conductor: \(3.36688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 725,\ (0:\ ),\ 0.856 - 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.047711343 - 0.2917461617i\)
\(L(\frac12)\) \(\approx\) \(1.047711343 - 0.2917461617i\)
\(L(1)\) \(\approx\) \(0.8382793500 - 0.2498441550i\)
\(L(1)\) \(\approx\) \(0.8382793500 - 0.2498441550i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.0448 - 0.998i)T \)
3 \( 1 + (-0.858 + 0.512i)T \)
7 \( 1 + (0.974 - 0.222i)T \)
11 \( 1 + (0.880 - 0.473i)T \)
13 \( 1 + (0.178 + 0.983i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.512 + 0.858i)T \)
23 \( 1 + (0.266 + 0.963i)T \)
31 \( 1 + (-0.351 - 0.936i)T \)
37 \( 1 + (-0.473 + 0.880i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.134 + 0.990i)T \)
53 \( 1 + (0.834 + 0.550i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.919 + 0.393i)T \)
67 \( 1 + (0.990 - 0.134i)T \)
71 \( 1 + (-0.134 + 0.990i)T \)
73 \( 1 + (-0.963 + 0.266i)T \)
79 \( 1 + (0.722 - 0.691i)T \)
83 \( 1 + (0.512 - 0.858i)T \)
89 \( 1 + (0.998 - 0.0448i)T \)
97 \( 1 + (0.995 - 0.0896i)T \)
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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

−22.8440216036207530062249539219, −22.03098133438606993014825985562, −21.38942911462700747369360509659, −19.997802591313787522254105724359, −19.09362786643815994882214489814, −18.155349614251875482963565755968, −17.62451670156806444257491444716, −17.08452677494728396067544417436, −16.253517397919784433924928348177, −15.10405841640564072374124887658, −14.703475476200475289332180569166, −13.567326355214168034448085834499, −12.72012986443563151369480157349, −12.03425134142540090831020508820, −10.87310777469913935573270927234, −10.1705640645797190334426834249, −8.74631995031627610595851396198, −8.18805445442781696523859092089, −7.12190900897111821753226497411, −6.50736874302058525673420162151, −5.46506845823127169796251850371, −4.88483001875187343228862387379, −3.83837979218535262101718058786, −1.95086177752279775486584708739, −0.82383968621410295750247031948, 0.99910176127197895618353627061, 1.852143093430507515838832163972, 3.4835167439654026141429859854, 4.19085743725098163178536271490, 5.010489479386922201117612561909, 5.922666016305479667191023753566, 7.16355358279048754120797290479, 8.42243170857001565676519095258, 9.30143346560465976654023571923, 10.04032452987450830264007165302, 11.06087918720311087095509690735, 11.60248010482478156067610487586, 12.020902175945838797975678932488, 13.32484762145421157885982125811, 14.20536545343981395201518997938, 14.8608837819386816346639308685, 16.23250127084007987383047193958, 17.00692416778582310103973718527, 17.53342123473551650587426738784, 18.55954521110484520867302587101, 19.082286586327894883664897844042, 20.3884569032988140485840657108, 20.89736374634323186049756799673, 21.629038755465097136380935391671, 22.229110124934611508596960532963

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