Properties

Degree 1
Conductor 163
Sign $-0.995 + 0.0967i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.211 − 0.977i)2-s + (−0.431 + 0.902i)3-s + (−0.910 + 0.413i)4-s + (−0.0581 − 0.998i)5-s + (0.973 + 0.230i)6-s + (−0.135 + 0.990i)7-s + (0.597 + 0.802i)8-s + (−0.627 − 0.778i)9-s + (−0.963 + 0.268i)10-s + (−0.565 − 0.824i)11-s + (0.0193 − 0.999i)12-s + (−0.993 − 0.116i)13-s + (0.996 − 0.0774i)14-s + (0.925 + 0.378i)15-s + (0.657 − 0.753i)16-s + (0.396 − 0.918i)17-s + ⋯
L(s,χ)  = 1  + (−0.211 − 0.977i)2-s + (−0.431 + 0.902i)3-s + (−0.910 + 0.413i)4-s + (−0.0581 − 0.998i)5-s + (0.973 + 0.230i)6-s + (−0.135 + 0.990i)7-s + (0.597 + 0.802i)8-s + (−0.627 − 0.778i)9-s + (−0.963 + 0.268i)10-s + (−0.565 − 0.824i)11-s + (0.0193 − 0.999i)12-s + (−0.993 − 0.116i)13-s + (0.996 − 0.0774i)14-s + (0.925 + 0.378i)15-s + (0.657 − 0.753i)16-s + (0.396 − 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.995 + 0.0967i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.995 + 0.0967i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(163\)
\( \varepsilon \)  =  $-0.995 + 0.0967i$
motivic weight  =  \(0\)
character  :  $\chi_{163} (15, \cdot )$
Sato-Tate  :  $\mu(81)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 163,\ (0:\ ),\ -0.995 + 0.0967i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.009559006235 - 0.1970541124i$
$L(\frac12,\chi)$  $\approx$  $0.009559006235 - 0.1970541124i$
$L(\chi,1)$  $\approx$  0.4729680619 - 0.1959234868i
$L(1,\chi)$  $\approx$  0.4729680619 - 0.1959234868i

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

−28.089361362819686835548360529990, −27.09385511460866101624850072489, −25.89671691823086543775982989912, −25.67465689675849228028367057716, −24.06237784974922740416141712058, −23.63810921524559705706967541111, −22.730027953590236560905987609328, −21.978859516475263021664067580618, −19.99024456881047605960563801912, −19.06122714718024723879555015202, −18.25442498427316308636733383051, −17.29882684260584254590067031822, −16.704284584282601724282509734578, −15.13484593583778255502420049848, −14.37911571980194473731099219471, −13.34720018939171388585017122992, −12.35633226918089158436302274448, −10.66967543669672657694978270782, −10.04437147779619727513938078291, −8.149097582502755392687470131245, −7.25863241702297937372544872308, −6.72953739247123583792626115360, −5.48545456882722443428828709468, −4.0152957201300781120093814923, −2.0041738734647651631040566976, 0.17569631184797394753752991580, 2.317112633902251743132067758737, 3.67772011084996094404951628289, 4.98167315711602971306532349798, 5.61960073434098495760794008177, 8.08214112868098353589049289406, 9.08151719573995058810915866114, 9.75740129946545155353413156905, 11.00042020514561224933672879182, 12.01419238855072630572755848500, 12.65189549922865197383489374331, 14.07239860356591047626702633295, 15.4824330169086817977720690964, 16.45618390102795678024066978866, 17.314076370606872978427547787369, 18.45636691403405445473344517757, 19.56220790793336051012943473716, 20.56750215933777805168337878176, 21.41991718544391704777598117061, 21.931452821525194593927431080546, 23.0719466738881778431207405515, 24.19790925915494675265802437531, 25.52057438334811095434049182574, 26.686409397478142416139135966709, 27.5951838535031393612381462253

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