Properties

Degree 1
Conductor 103
Sign $-0.451 - 0.892i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.451 - 0.892i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.451 - 0.892i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(103\)
\( \varepsilon \)  =  $-0.451 - 0.892i$
motivic weight  =  \(0\)
character  :  $\chi_{103} (79, \cdot )$
Sato-Tate  :  $\mu(17)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 103,\ (0:\ ),\ -0.451 - 0.892i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6414095373 - 1.043757616i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6414095373 - 1.043757616i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9290936898 - 0.8062977315i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9290936898 - 0.8062977315i\)

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

−30.72327223756490043193038263904, −28.5689217178733066185028031164, −28.083159968338702365746393690311, −26.72951476885820626008601961795, −25.93905213302362669275999899559, −24.801043330844296726424318352938, −24.58736469834459100709076172621, −22.80300664444085569652790202301, −21.85225615991728571193783265597, −20.95231546204836080225265976128, −19.71179250337833967559014696157, −18.38939697594308983854963652858, −17.170144095031605040506833286822, −16.06878428423959331394708545771, −15.41289016328034244740409922991, −14.23472116515258916382400978026, −13.27698589161444175638664880652, −12.0638563389911065153218577289, −9.668092698827119936418462684364, −9.26927253562428477413383036333, −8.22337162616655734280428043552, −6.68787038580889016350932841399, −5.10001762385334328491275422111, −4.41537065914587702624824559298, −2.42330579546674959258965043312, 1.32898461908712002210946281177, 2.927364418974554271016173579353, 3.65732091122138563384018258622, 5.85068177795455877368550765301, 7.30398195085041592406396758454, 8.59225553753054881208290670673, 9.992122847528716510423518344217, 10.79213281470230625163242473411, 12.24274666849762719924006853803, 13.61804117351521481132297144099, 13.82615338599251623249253323713, 15.15125165294285866683585442793, 17.28054535611644199711899121634, 18.13993806205810113079924119398, 19.222159877044717006215366781876, 19.85395021228238067113792033279, 20.971775968540140297201509165094, 22.089999610862704749780248750719, 23.091356866538952243998563129830, 24.1621599628121149359163197275, 25.56483115149262440061295580576, 26.58135225263296552668352929152, 27.15531505094872791824007374079, 29.05450311624981119015408701457, 29.61771105874748412147958114790

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