Properties

Degree 1
Conductor $ 5^{2} $
Sign $0.248 + 0.968i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.248 + 0.968i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.248 + 0.968i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(25\)    =    \(5^{2}\)
\( \varepsilon \)  =  $0.248 + 0.968i$
motivic weight  =  \(0\)
character  :  $\chi_{25} (17, \cdot )$
Sato-Tate  :  $\mu(20)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 25,\ (1:\ ),\ 0.248 + 0.968i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3058709774 + 0.2372578502i$
$L(\frac12,\chi)$  $\approx$  $0.3058709774 + 0.2372578502i$
$L(\chi,1)$  $\approx$  0.4848512120 + 0.02760602250i
$L(1,\chi)$  $\approx$  0.4848512120 + 0.02760602250i

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

−37.60513145579877694390295059206, −36.29671920187941390681171293440, −35.124733557385188846998445571937, −34.02714081331372458594774403030, −33.14658241292956816995184174647, −31.96836609657264176240906051068, −29.651770418372795235961329998391, −28.654241666500588188044863021371, −27.16399880243125666591279078673, −26.5403620998003768679336352229, −24.640176484335156966507978799951, −23.49009302667954246447631091283, −22.479158317839920244834698576315, −20.497444812080845056448426103645, −18.68394949104805753415221212631, −17.37530766536392588808458507484, −16.46563996392816622255218531425, −15.13970570819968589009419891233, −13.263967648471832601029059351472, −10.96026688678815330242805263860, −9.95131118427465779841704816020, −7.81910086527958365643836925181, −6.26631616703937676868385893554, −4.70026480711542061894887985631, −0.39644555033453620608135353368, 2.14844988942813312364228150697, 4.90189089629268277047822372681, 7.11896130669450051008926366492, 9.07754959898039399954425069144, 10.740664138904519704033883926859, 11.97914172099989670851315075138, 13.05600052478992183658929999412, 15.70375978327536940787543772921, 17.30882065413628364207598943123, 18.282761273156686249270877059514, 19.46763958190183144695204980447, 21.372830439383272431502404615973, 22.220954773652521227645302647510, 23.85584942075733104719855047068, 25.52406529140324674648764539033, 27.09540874190338327240252041075, 28.47703090611749816448521120021, 28.88819627657530172206580130683, 30.4591037814164988461560766139, 31.547321921431999773449740222682, 33.7321194257003710551920296391, 34.78142326309883418651212081123, 35.80960056672587348378694907764, 37.06307838299446252015000881138, 38.560399837280829116862981053247

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