Properties

Degree 1
Conductor 367
Sign $0.105 - 0.994i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.988 − 0.153i)2-s + (0.423 − 0.905i)3-s + (0.952 + 0.304i)4-s + (0.870 − 0.492i)5-s + (−0.558 + 0.829i)6-s + (0.679 + 0.733i)7-s + (−0.894 − 0.447i)8-s + (−0.640 − 0.767i)9-s + (−0.935 + 0.352i)10-s + (0.751 + 0.660i)11-s + (0.679 − 0.733i)12-s + (−0.640 − 0.767i)13-s + (−0.558 − 0.829i)14-s + (−0.0771 − 0.997i)15-s + (0.815 + 0.579i)16-s + (−0.998 + 0.0514i)17-s + ⋯
L(s,χ)  = 1  + (−0.988 − 0.153i)2-s + (0.423 − 0.905i)3-s + (0.952 + 0.304i)4-s + (0.870 − 0.492i)5-s + (−0.558 + 0.829i)6-s + (0.679 + 0.733i)7-s + (−0.894 − 0.447i)8-s + (−0.640 − 0.767i)9-s + (−0.935 + 0.352i)10-s + (0.751 + 0.660i)11-s + (0.679 − 0.733i)12-s + (−0.640 − 0.767i)13-s + (−0.558 − 0.829i)14-s + (−0.0771 − 0.997i)15-s + (0.815 + 0.579i)16-s + (−0.998 + 0.0514i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.105 - 0.994i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.105 - 0.994i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(367\)
\( \varepsilon \)  =  $0.105 - 0.994i$
motivic weight  =  \(0\)
character  :  $\chi_{367} (135, \cdot )$
Sato-Tate  :  $\mu(61)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 367,\ (0:\ ),\ 0.105 - 0.994i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8716361485 - 0.7837791557i$
$L(\frac12,\chi)$  $\approx$  $0.8716361485 - 0.7837791557i$
$L(\chi,1)$  $\approx$  0.8844854118 - 0.4113383007i
$L(1,\chi)$  $\approx$  0.8844854118 - 0.4113383007i

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

−25.00326769947323590307058494112, −24.461034522007823571109607377678, −23.13652566079728179488959505225, −21.821632364991276152641980762627, −21.32955699308834074666492144642, −20.44368552759393186992468286029, −19.56515235555549551823901218375, −18.78047496552558646479185002944, −17.46151171821882455432930494649, −17.07495707911087514914813222444, −16.23312633449401316678434326063, −14.98541418822998325073697580306, −14.38093525352286638900733602014, −13.64099997932543745775506327367, −11.715625135551366099336108549283, −10.83948572200774822149232291464, −10.27659866254238548269165551282, −9.237873860136522866347651115197, −8.660280443238974622015930896966, −7.37375010602504486713997487440, −6.44551235452473388988864834710, −5.23305595227533296515867159184, −3.91451725912013643513554682829, −2.58639972743269346666677648419, −1.54081066697264151745080796516, 1.000088094981471220145344014819, 2.1282168796612302240077994331, 2.65859579931881735712034781876, 4.76441366603798692641464585795, 6.150725249410458205219502264250, 6.902654418701774822537821467608, 8.09133531764933562443993445160, 8.85823671748895490132133768251, 9.461064613345531907388532984225, 10.73929107431478451921967765567, 11.94528971438949335783854533512, 12.51823950733288575966992789861, 13.536455481441124056810546999698, 14.80857111931679360654423413799, 15.41798989044033786478063861040, 17.14024663282944076956662152335, 17.49217588457891036430568617867, 18.08828655998040184916826639384, 19.19363418999492329483829554265, 19.92140141412673853298926753440, 20.70664076487991954845584362554, 21.51781593977633741617583805133, 22.651167587247843466874766493463, 24.37527916588718284245201437846, 24.53581731191964096128984636520

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