Properties

Degree 1
Conductor $ 3 \cdot 5^{2} $
Sign $-0.425 + 0.904i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s − 7-s + (−0.809 + 0.587i)8-s + (−0.309 + 0.951i)11-s + (−0.309 − 0.951i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)22-s + (0.309 − 0.951i)23-s − 26-s + (0.809 + 0.587i)28-s + (0.809 + 0.587i)29-s + ⋯
L(s,χ)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s − 7-s + (−0.809 + 0.587i)8-s + (−0.309 + 0.951i)11-s + (−0.309 − 0.951i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)22-s + (0.309 − 0.951i)23-s − 26-s + (0.809 + 0.587i)28-s + (0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.425 + 0.904i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.425 + 0.904i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(75\)    =    \(3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.425 + 0.904i$
motivic weight  =  \(0\)
character  :  $\chi_{75} (14, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 75,\ (1:\ ),\ -0.425 + 0.904i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.04420616673 - 0.06965777263i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.04420616673 - 0.06965777263i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6357777996 - 0.3495218031i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6357777996 - 0.3495218031i\)

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

−31.913920914084507512144951120689, −31.25889403861981853241173415455, −29.748695684589697493594639298151, −28.71733134598476622758510560636, −27.17711782242439620320103706788, −26.31332728536689014915398591631, −25.4147515720087616866961495343, −24.220207542718535533050582807308, −23.3624945114136360745236574117, −22.15492722365398055084108309145, −21.37765446822366796281369381270, −19.53427970298240066701174836940, −18.52680915430992150165182765777, −17.103074852765470190002252635421, −16.1967993687067881097940946607, −15.25190626869013423332180504305, −13.7743420398259030772506897242, −13.061382823037689603893063710, −11.52462731851607203037641272075, −9.650390495473313599727607380685, −8.60660878905825794819903096350, −7.04333535542943088126746612175, −6.09694482112194161064627066633, −4.561584291274186832023498064691, −3.05447251532235183969810889736, 0.034781810516460489136878182495, 2.182683380260347232114542670249, 3.59187693176354839218970107253, 5.05376889430521604329438206235, 6.60456811334367194005075113597, 8.55699999068663908463874167931, 9.938993230513482493878735029520, 10.719379852575104641996969188952, 12.56187923476548048403030568827, 12.83706008157947641544221726297, 14.48364794039986402532465117774, 15.596930751746574346021973010962, 17.29462376070722159238064850937, 18.4357033074746015941940499634, 19.63391163268307454259972031309, 20.34967719885340706729766260319, 21.69246374143386805409167345286, 22.6781705935593212532115021885, 23.42447629278815933291668625798, 25.00817064561824613083744452112, 26.23143522492909115675111228278, 27.4477017023164689029032731087, 28.517708018997239150427243032111, 29.29002048452261153870715910021, 30.38342996382534435769374429951

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