Properties

Degree 1
Conductor 193
Sign $-0.805 + 0.592i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.980 − 0.195i)2-s + (−0.382 − 0.923i)3-s + (0.923 − 0.382i)4-s + (−0.0980 − 0.995i)5-s + (−0.555 − 0.831i)6-s + (−0.707 + 0.707i)7-s + (0.831 − 0.555i)8-s + (−0.707 + 0.707i)9-s + (−0.290 − 0.956i)10-s + (−0.773 + 0.634i)11-s + (−0.707 − 0.707i)12-s + (−0.956 − 0.290i)13-s + (−0.555 + 0.831i)14-s + (−0.881 + 0.471i)15-s + (0.707 − 0.707i)16-s + (0.0980 − 0.995i)17-s + ⋯
L(s,χ)  = 1  + (0.980 − 0.195i)2-s + (−0.382 − 0.923i)3-s + (0.923 − 0.382i)4-s + (−0.0980 − 0.995i)5-s + (−0.555 − 0.831i)6-s + (−0.707 + 0.707i)7-s + (0.831 − 0.555i)8-s + (−0.707 + 0.707i)9-s + (−0.290 − 0.956i)10-s + (−0.773 + 0.634i)11-s + (−0.707 − 0.707i)12-s + (−0.956 − 0.290i)13-s + (−0.555 + 0.831i)14-s + (−0.881 + 0.471i)15-s + (0.707 − 0.707i)16-s + (0.0980 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.805 + 0.592i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 193 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.805 + 0.592i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(193\)
\( \varepsilon \)  =  $-0.805 + 0.592i$
motivic weight  =  \(0\)
character  :  $\chi_{193} (13, \cdot )$
Sato-Tate  :  $\mu(64)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 193,\ (1:\ ),\ -0.805 + 0.592i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.2984908036 - 0.9089094429i$
$L(\frac12,\chi)$  $\approx$  $-0.2984908036 - 0.9089094429i$
$L(\chi,1)$  $\approx$  0.9525197868 - 0.6811383903i
$L(1,\chi)$  $\approx$  0.9525197868 - 0.6811383903i

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

−27.060997744530302620678236489206, −26.23951941307815465626384770542, −25.83543795568845210344012285549, −24.19798232726154782183699798417, −23.21227008891130556955656074864, −22.74068835166800028920628402171, −21.683004138599170660358911219508, −21.231244488380042118360804579335, −19.8768808702370786907226489709, −18.99277566643494770992390639722, −17.16969392302198885978354108936, −16.68618760071411910658247771030, −15.3827165887684876255059869712, −14.9505488475633173914429119642, −13.79430637172758502040537680430, −12.75973130508134331043316118140, −11.387920887894137925078473036666, −10.691127310919233390544622852154, −9.84836250951785363719523383053, −7.953926324120837812625791598804, −6.69971874449615413502670998183, −5.925358300955184310278910537989, −4.56401159067398330193451183068, −3.5786508191605689425409418809, −2.65613603942167103173204475949, 0.223860552760128035962687211968, 1.896096523629936256842490642503, 2.93099061531307019660138049095, 4.87662568904913329348296651288, 5.4024718429132374188134067990, 6.7123773504179489032132520871, 7.67423786164002973998764415254, 9.15898738737062000263932071028, 10.58024710371510857343599745833, 11.90696025761676198597135113454, 12.74478281203075419183458785173, 12.8549333647298751743050888429, 14.28642304544564684189673678329, 15.52470827854925652171983984607, 16.38431299350438630695762304671, 17.4452811423735313761573315143, 18.80664868622388427831225578764, 19.614781021273222106933280855815, 20.51643611800039892992906996459, 21.57992026641237961563246692190, 22.70391114840659654145325001282, 23.29621176029394648030771879034, 24.29012335194581062093605727164, 25.0197747461584635386057178658, 25.60895085769563870774631838108

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