Properties

Degree 1
Conductor $ 5 \cdot 47 $
Sign $0.467 + 0.883i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.979 − 0.203i)2-s + (0.730 + 0.682i)3-s + (0.917 + 0.398i)4-s + (−0.576 − 0.816i)6-s + (0.631 + 0.775i)7-s + (−0.816 − 0.576i)8-s + (0.0682 + 0.997i)9-s + (0.334 − 0.942i)11-s + (0.398 + 0.917i)12-s + (0.136 + 0.990i)13-s + (−0.460 − 0.887i)14-s + (0.682 + 0.730i)16-s + (−0.942 + 0.334i)17-s + (0.136 − 0.990i)18-s + (0.962 − 0.269i)19-s + ⋯
L(s,χ)  = 1  + (−0.979 − 0.203i)2-s + (0.730 + 0.682i)3-s + (0.917 + 0.398i)4-s + (−0.576 − 0.816i)6-s + (0.631 + 0.775i)7-s + (−0.816 − 0.576i)8-s + (0.0682 + 0.997i)9-s + (0.334 − 0.942i)11-s + (0.398 + 0.917i)12-s + (0.136 + 0.990i)13-s + (−0.460 − 0.887i)14-s + (0.682 + 0.730i)16-s + (−0.942 + 0.334i)17-s + (0.136 − 0.990i)18-s + (0.962 − 0.269i)19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.467 + 0.883i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.467 + 0.883i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(235\)    =    \(5 \cdot 47\)
\( \varepsilon \)  =  $0.467 + 0.883i$
motivic weight  =  \(0\)
character  :  $\chi_{235} (163, \cdot )$
Sato-Tate  :  $\mu(92)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 235,\ (0:\ ),\ 0.467 + 0.883i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9109463041 + 0.5485753514i$
$L(\frac12,\chi)$  $\approx$  $0.9109463041 + 0.5485753514i$
$L(\chi,1)$  $\approx$  0.9126707556 + 0.2702412378i
$L(1,\chi)$  $\approx$  0.9126707556 + 0.2702412378i

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

−26.10986731965599451331001495800, −25.19533063612334070436277969094, −24.58419998330220347832784476179, −23.70456977918363851787182688240, −22.65534674921278895360028106475, −20.80308504069335079935132643986, −20.30404854264280650181800695372, −19.68742779073149360031038725140, −18.45228997038979563735755405575, −17.77151845109183634973912294775, −17.09394584341980364672779573762, −15.61168187080579578896471826503, −14.85195335488119506161758280522, −13.87742772020474922491837288234, −12.69722103347407725218550575881, −11.5396263393230809854837749140, −10.45458957316920459364600876427, −9.36504999916986488981699194133, −8.448295083224670596797182149334, −7.332525442287709020631065926467, −7.00561970306532523075965651747, −5.31720949991564406817443636204, −3.53457862482183289145784583139, −2.1235327467528496258734937916, −1.06574133709649429827391818457, 1.70012405867804326736777369417, 2.77585072593347558725442620202, 4.00230991027414273474401828828, 5.53268823543418598799248317030, 6.99279802238909292280429954292, 8.262642115241213942742769693, 8.93445983425954493595065441967, 9.592996189109899458244547714135, 11.15687293956368245793106974943, 11.38167732580577263090834620288, 13.06004505712229898420906033590, 14.3421864378527935178369716019, 15.2328285025732715556258633779, 16.13669474569630980496229123565, 16.93921473114292970047311880070, 18.21327115159491405902769181797, 19.01301026385026721872889828196, 19.80103793574045632481817446938, 20.86543555922747723319960107551, 21.476610317679170074116807126297, 22.25087740205995083243538063540, 24.420561749872071839738131265332, 24.50966530057036276225718469660, 25.8272503431199552291071102815, 26.55063158633571969565669376421

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