Properties

Degree 1
Conductor $ 5^{2} $
Sign $0.187 + 0.982i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.951 + 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s + i·7-s + (0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (−0.951 + 0.309i)12-s + (0.951 − 0.309i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s i·18-s + (0.809 − 0.587i)19-s + ⋯
L(s,χ)  = 1  + (0.951 + 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s + i·7-s + (0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (−0.951 + 0.309i)12-s + (0.951 − 0.309i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s i·18-s + (0.809 − 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.187 + 0.982i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.187 + 0.982i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(25\)    =    \(5^{2}\)
\( \varepsilon \)  =  $0.187 + 0.982i$
motivic weight  =  \(0\)
character  :  $\chi_{25} (2, \cdot )$
Sato-Tate  :  $\mu(20)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 25,\ (1:\ ),\ 0.187 + 0.982i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.482218465 + 1.226197754i$
$L(\frac12,\chi)$  $\approx$  $1.482218465 + 1.226197754i$
$L(\chi,1)$  $\approx$  1.378802374 + 0.7129109418i
$L(1,\chi)$  $\approx$  1.378802374 + 0.7129109418i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−38.09866824913236886118927208397, −36.470756307170771026140700066559, −35.25730937377627798835378689423, −33.63428978147748324985848301743, −32.94093397731409517051495856432, −30.994475497945596764249261700722, −30.27505466527500294819961545390, −29.10355236216309161927950165122, −28.00979912702797634246984899424, −25.74427898855974964160548641100, −24.270126704651321674008307671680, −23.31070693443765994536139747628, −22.37283659680972527442878489317, −20.58085192224977800941300201367, −19.404237458897755627662812147778, −17.67324902540025821502366222951, −16.11990570142744652770830929631, −14.167424483808897243122084319674, −13.07143163162304960966022768946, −11.74692761578520974197903825592, −10.4178092774473160463202280021, −7.39593930121967567884613681378, −6.0762064126980081679370655263, −4.159560687065339514036455869063, −1.58220676000882298958522643392, 3.299412304185740963460154106670, 5.12731131686940929145413272001, 6.3107254305470667701549426079, 8.78700865446590678715430717171, 11.01035637784640690626539889546, 12.09271183978261830717139328509, 13.93504548429567351049649728926, 15.5407910552972031602988945058, 16.243900265057743914117467792210, 18.01987117849817655242877267313, 20.326983525217915866082945291589, 21.681153704098747260886540395419, 22.35576466016837459450057651849, 23.80309499718615968431776583302, 25.14731898512247250354570849926, 26.63347539267659867738715781851, 28.20148379648412894448495556373, 29.3956505188558257526594211547, 31.01676168048359971116478493543, 32.19829178912301325466792523076, 33.12199662141554282205170977353, 34.430937675677178326012078771082, 35.148544981062632168677880280817, 37.74583052648439184978635236180, 38.4452568534993322207135798615

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