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Properties

Label 1-5e2-25.13-r1-0-0
Degree $1$
Conductor $25$
Sign $0.187 - 0.982i$
Analytic cond. $2.68662$
Root an. cond. $2.68662$
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  + (0.951 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.187 - 0.982i$
Analytic conductor: \(2.68662\)
Root analytic conductor: \(2.68662\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 25,\ (1:\ ),\ 0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.482218465 - 1.226197754i\)
\(L(\frac12)\) \(\approx\) \(1.482218465 - 1.226197754i\)
\(L(1)\) \(\approx\) \(1.378802374 - 0.7129109418i\)
\(L(1)\) \(\approx\) \(1.378802374 - 0.7129109418i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
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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

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

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