Properties

Degree 1
Conductor $ 3 \cdot 31 $
Sign $-0.798 + 0.602i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + 4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s − 23-s + ⋯
L(s,χ)  = 1  − 2-s + 4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + 16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s − 23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.798 + 0.602i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 93 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.798 + 0.602i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(93\)    =    \(3 \cdot 31\)
\( \varepsilon \)  =  $-0.798 + 0.602i$
motivic weight  =  \(0\)
character  :  $\chi_{93} (56, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 93,\ (1:\ ),\ -0.798 + 0.602i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2187946884 + 0.6535815738i$
$L(\frac12,\chi)$  $\approx$  $0.2187946884 + 0.6535815738i$
$L(\chi,1)$  $\approx$  0.5998185980 + 0.2543948967i
$L(1,\chi)$  $\approx$  0.5998185980 + 0.2543948967i

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

−29.475410375005417585225016511420, −28.52313277506258976478026834739, −27.62571886886008229027487059326, −26.394147422577901977444721248481, −25.75431945398809797658750456109, −24.36612138195045829147271501806, −23.85394473649940596697579361194, −21.900250768289025398834765188608, −20.94646243468254286725380218352, −19.77588849659159618513660650886, −19.15067591117725946941388695039, −17.53647542181117599512471385816, −16.7928525355098098733271371492, −16.12093732207766824924140233675, −14.39861202328300879342834380921, −13.092501771856424950036710487692, −11.80561296362779283600848189724, −10.46957465128769311664786818832, −9.42554341400735993895765954899, −8.476808404630460290446370550478, −7.02524979239792192606006412639, −5.84409865594134560183811074547, −3.87096759474726129300922858420, −1.87011198980607890160051341914, −0.40894082763904199520033858972, 1.956329213256190470949270906826, 3.12192735342213020993997043611, 5.68928622465211481590440681007, 6.757263055718111821850103388064, 7.972031561536890525205796881203, 9.57932215995710996408027729630, 10.06468058817011563191786654260, 11.5734355426482246358477978732, 12.62626826568755103671235001760, 14.54841069276421502385209124688, 15.32615706099759062191566336113, 16.64376921882902215264673080496, 17.85315863067270931424354681821, 18.51491681407989166034553775908, 19.57343096733988701567623551160, 20.73484630930158245903649988903, 21.99995756341492897909731917803, 22.89860500915197267879744675334, 24.77425511440022684419211447064, 25.34820439915248779864084826172, 26.18736191232762979190970622458, 27.43691231289964944304179371542, 28.184633342001237183041919211158, 29.500882544493381868090528352836, 29.96225145342679722174738312461

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