Properties

Degree 1
Conductor 103
Sign $-0.0945 - 0.995i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.273 − 0.961i)2-s + (−0.602 + 0.798i)3-s + (−0.850 + 0.526i)4-s + (−0.982 + 0.183i)5-s + (0.932 + 0.361i)6-s + (0.739 − 0.673i)7-s + (0.739 + 0.673i)8-s + (−0.273 − 0.961i)9-s + (0.445 + 0.895i)10-s + (−0.273 − 0.961i)11-s + (0.0922 − 0.995i)12-s + (0.739 − 0.673i)13-s + (−0.850 − 0.526i)14-s + (0.445 − 0.895i)15-s + (0.445 − 0.895i)16-s + (0.932 − 0.361i)17-s + ⋯
L(s,χ)  = 1  + (−0.273 − 0.961i)2-s + (−0.602 + 0.798i)3-s + (−0.850 + 0.526i)4-s + (−0.982 + 0.183i)5-s + (0.932 + 0.361i)6-s + (0.739 − 0.673i)7-s + (0.739 + 0.673i)8-s + (−0.273 − 0.961i)9-s + (0.445 + 0.895i)10-s + (−0.273 − 0.961i)11-s + (0.0922 − 0.995i)12-s + (0.739 − 0.673i)13-s + (−0.850 − 0.526i)14-s + (0.445 − 0.895i)15-s + (0.445 − 0.895i)16-s + (0.932 − 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.0945 - 0.995i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.0945 - 0.995i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(103\)
\( \varepsilon \)  =  $-0.0945 - 0.995i$
motivic weight  =  \(0\)
character  :  $\chi_{103} (14, \cdot )$
Sato-Tate  :  $\mu(17)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 103,\ (0:\ ),\ -0.0945 - 0.995i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.3680853605 - 0.4047061827i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.3680853605 - 0.4047061827i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5803520530 - 0.2634950716i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5803520530 - 0.2634950716i\)

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

−30.438732262160413331050383444458, −28.46241283658473347400031301464, −28.11357717121096830793305629296, −27.12563233379177789094465756072, −25.68625817663424939950865845019, −24.84547628345352043493148677480, −23.69739575276430011634164036550, −23.407128498179876799283912306734, −22.23334722521314797523968460031, −20.5566262367511788379199604543, −18.84609449674328882850259338092, −18.6405517222003402067262783810, −17.31889155926958373934091799374, −16.394305338233348360705266245129, −15.262027787491668103214941930861, −14.26161085037561859558625060743, −12.7221091221317447608607807425, −11.86549676702328262260823622820, −10.47490328561496163964664819403, −8.58022947910017724121685748855, −7.887227140110030224595771011519, −6.75567891048689087875411320660, −5.48438021180972694715307021182, −4.33238651802080697586618220105, −1.57406837601252276027012563662, 0.729797320171090458090053493412, 3.28459689224164605660212581605, 4.13140272591436135023312828453, 5.44006625945478897717697289959, 7.628077823567219585299176065042, 8.69703986971049732990259739641, 10.21834562549125920550892107832, 11.15546465350001730427348694657, 11.57455602266361302445892078356, 13.13950628786335529269623028916, 14.53957575416807716361066506762, 15.86970369743994690244922064125, 16.92357246004157153775074158777, 17.98558790265640432740333019747, 19.10443253936322796367840733207, 20.333931707625679876345909689716, 21.02909703242086354366244080843, 22.10723748536868670685258416084, 23.20475520987868955582293023998, 23.78507825309527445452005548161, 26.02340154599798791584093482224, 26.8717682429577375104432639806, 27.63605268610028964243106315774, 28.14499925213479408145209998072, 29.68951112768521612645258700537

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