Properties

Degree 1
Conductor 367
Sign $-0.432 + 0.901i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.423 − 0.905i)2-s + (−0.558 − 0.829i)3-s + (−0.640 − 0.767i)4-s + (0.916 − 0.400i)5-s + (−0.988 + 0.153i)6-s + (−0.279 − 0.960i)7-s + (−0.967 + 0.254i)8-s + (−0.376 + 0.926i)9-s + (0.0257 − 0.999i)10-s + (−0.998 + 0.0514i)11-s + (−0.279 + 0.960i)12-s + (−0.376 + 0.926i)13-s + (−0.988 − 0.153i)14-s + (−0.843 − 0.536i)15-s + (−0.179 + 0.983i)16-s + (−0.784 + 0.620i)17-s + ⋯
L(s,χ)  = 1  + (0.423 − 0.905i)2-s + (−0.558 − 0.829i)3-s + (−0.640 − 0.767i)4-s + (0.916 − 0.400i)5-s + (−0.988 + 0.153i)6-s + (−0.279 − 0.960i)7-s + (−0.967 + 0.254i)8-s + (−0.376 + 0.926i)9-s + (0.0257 − 0.999i)10-s + (−0.998 + 0.0514i)11-s + (−0.279 + 0.960i)12-s + (−0.376 + 0.926i)13-s + (−0.988 − 0.153i)14-s + (−0.843 − 0.536i)15-s + (−0.179 + 0.983i)16-s + (−0.784 + 0.620i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.432 + 0.901i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.432 + 0.901i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(367\)
\( \varepsilon \)  =  $-0.432 + 0.901i$
motivic weight  =  \(0\)
character  :  $\chi_{367} (74, \cdot )$
Sato-Tate  :  $\mu(61)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 367,\ (0:\ ),\ -0.432 + 0.901i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.3890304113 - 0.6181751472i$
$L(\frac12,\chi)$  $\approx$  $-0.3890304113 - 0.6181751472i$
$L(\chi,1)$  $\approx$  0.4434962421 - 0.7452492403i
$L(1,\chi)$  $\approx$  0.4434962421 - 0.7452492403i

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

−25.39839934462747456772437046656, −24.471929587002053965000922415918, −23.346263705879838869196620375692, −22.497753275737819037957763336223, −21.99234577508149203364693786341, −21.29757197983065916128819312327, −20.43136722722857583738021429487, −18.5082172175601628060182275734, −18.05180801443593514073332524767, −17.18056242336291283772314780063, −16.25219113788680305961060268517, −15.354809975138177966654713238085, −14.95156349751145739266209916200, −13.70175218360560145467403918254, −12.828173548595433242274137579743, −11.855013738716217452452480284048, −10.554138079089517795833990453418, −9.65910835390875839242514037660, −8.86683521199135025479365433618, −7.56190172576919472006646265436, −6.28675172927241627047030103172, −5.53998705785942367525224447662, −5.08513235630181955098210967832, −3.507662997794837277718704980638, −2.54651997407242489699617674364, 0.38523127796195604892057399301, 1.787840380631932021797259904199, 2.51917737998751355654957514968, 4.27187319160746117121491986852, 5.13325139158011415460424131608, 6.18276821542426332262675541233, 7.08537903351691912477565587801, 8.56949488372374000020347294658, 9.69737632613409357367894993321, 10.636115444344448728418921366139, 11.29604348497092824980295928368, 12.61960252087509304913692376964, 13.15974465193508296179452056871, 13.66771366138725510051429331258, 14.72971598814864764783110738859, 16.3767645805483302377558998878, 17.18386451866978488300836339995, 18.03114842380580348776832373601, 18.77192782138245224804296687961, 19.844211410803748571686589083847, 20.45149527196099199820913502826, 21.61710645363860868127419180689, 22.16951650847138391286817919085, 23.25684233342061723304937594551, 23.98424342631144083519845447388

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