Properties

Degree 1
Conductor $ 5 \cdot 47 $
Sign $-0.352 + 0.935i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.519 + 0.854i)2-s + (−0.816 + 0.576i)3-s + (−0.460 − 0.887i)4-s + (−0.0682 − 0.997i)6-s + (−0.269 + 0.962i)7-s + (0.997 + 0.0682i)8-s + (0.334 − 0.942i)9-s + (0.990 − 0.136i)11-s + (0.887 + 0.460i)12-s + (0.631 − 0.775i)13-s + (−0.682 − 0.730i)14-s + (−0.576 + 0.816i)16-s + (0.136 − 0.990i)17-s + (0.631 + 0.775i)18-s + (0.203 + 0.979i)19-s + ⋯
L(s,χ)  = 1  + (−0.519 + 0.854i)2-s + (−0.816 + 0.576i)3-s + (−0.460 − 0.887i)4-s + (−0.0682 − 0.997i)6-s + (−0.269 + 0.962i)7-s + (0.997 + 0.0682i)8-s + (0.334 − 0.942i)9-s + (0.990 − 0.136i)11-s + (0.887 + 0.460i)12-s + (0.631 − 0.775i)13-s + (−0.682 − 0.730i)14-s + (−0.576 + 0.816i)16-s + (0.136 − 0.990i)17-s + (0.631 + 0.775i)18-s + (0.203 + 0.979i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.352 + 0.935i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.352 + 0.935i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(235\)    =    \(5 \cdot 47\)
\( \varepsilon \)  =  $-0.352 + 0.935i$
motivic weight  =  \(0\)
character  :  $\chi_{235} (137, \cdot )$
Sato-Tate  :  $\mu(92)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 235,\ (0:\ ),\ -0.352 + 0.935i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.3885150821 + 0.5613961554i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.3885150821 + 0.5613961554i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5501745653 + 0.3834625461i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5501745653 + 0.3834625461i\)

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

−26.2499317049876745801475044147, −25.16042622445371388546631958768, −23.94450735146082083386699244340, −23.07644140034321778123334783227, −22.27229602927760022236131702621, −21.37711332798421091280628297267, −20.183539646697854234011154572543, −19.35123389152486271154720677164, −18.66168816211836387138694503821, −17.3840572647967292831882742370, −17.05798953289486144612762268982, −16.07884694304235820887538911169, −14.167167687049608875555002109896, −13.2625658973394160117531433993, −12.437433279891063782705294313385, −11.35724851607822983182037389603, −10.77634193508314284709387940960, −9.64495173554875747528673176524, −8.46198327479212476093713669226, −7.18285051531276977734632087301, −6.43331630050231562581835205616, −4.6335179416143137757577730974, −3.68392478156804583756568028834, −1.91595410028161732598545593208, −0.83081631544153292867790366458, 1.19561961075539917811674512388, 3.45853800390241943839006615598, 4.93164236392954505185183917701, 5.82701039867459883362593445818, 6.52901261226847390571543182732, 7.952463054827202636609612676566, 9.23046163208088186644588335816, 9.735334455411150013579927696382, 11.06120237208476715335321801380, 11.943860815271280410945681692777, 13.29349717729319783116610829511, 14.70355376112574658586529952123, 15.380972747320201626087335695298, 16.292547581668334049991118536147, 16.95241974078554168419168583910, 18.108962650739243656718127658053, 18.60405183083137252084227457081, 19.91635337511638599359805404044, 21.118964649524658760422050330223, 22.41628196859108083606797636833, 22.68956558244066338852788856339, 23.820876240053979803075742718139, 24.97386037895740198052150263390, 25.41616453489321074918885126477, 26.74753547979939106839141566508

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