Properties

Degree 1
Conductor 103
Sign $0.534 - 0.845i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.992 − 0.122i)2-s + (0.445 − 0.895i)3-s + (0.969 − 0.243i)4-s + (−0.908 + 0.417i)5-s + (0.332 − 0.943i)6-s + (−0.153 − 0.988i)7-s + (0.932 − 0.361i)8-s + (−0.602 − 0.798i)9-s + (−0.850 + 0.526i)10-s + (−0.389 + 0.920i)11-s + (0.213 − 0.976i)12-s + (0.932 + 0.361i)13-s + (−0.273 − 0.961i)14-s + (−0.0307 + 0.999i)15-s + (0.881 − 0.473i)16-s + (0.332 + 0.943i)17-s + ⋯
L(s,χ)  = 1  + (0.992 − 0.122i)2-s + (0.445 − 0.895i)3-s + (0.969 − 0.243i)4-s + (−0.908 + 0.417i)5-s + (0.332 − 0.943i)6-s + (−0.153 − 0.988i)7-s + (0.932 − 0.361i)8-s + (−0.602 − 0.798i)9-s + (−0.850 + 0.526i)10-s + (−0.389 + 0.920i)11-s + (0.213 − 0.976i)12-s + (0.932 + 0.361i)13-s + (−0.273 − 0.961i)14-s + (−0.0307 + 0.999i)15-s + (0.881 − 0.473i)16-s + (0.332 + 0.943i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.534 - 0.845i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.534 - 0.845i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(103\)
\( \varepsilon \)  =  $0.534 - 0.845i$
motivic weight  =  \(0\)
character  :  $\chi_{103} (2, \cdot )$
Sato-Tate  :  $\mu(51)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 103,\ (0:\ ),\ 0.534 - 0.845i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.553493494 - 0.8554101287i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.553493494 - 0.8554101287i\)
\(L(\chi,1)\)  \(\approx\)  \(1.624580755 - 0.5850154404i\)
\(L(1,\chi)\)  \(\approx\)  \(1.624580755 - 0.5850154404i\)

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

−30.42969146227313420495338842702, −28.85589866212727609835694870175, −27.947668083144742185937265181251, −26.888316359856823729767734990703, −25.60869454317352718539029920359, −24.83780153122764403851588979097, −23.60351331062531509453387889672, −22.65632260392697269183036654637, −21.58236386063364863053755021253, −20.83270040633168495220173957213, −19.850048822505462903737425072031, −18.71304606120376599123114441914, −16.4974681387312259991739395994, −15.918797993675631022708317446716, −15.18285671566740053689933576047, −14.02711972471929503570650324483, −12.77418089379019711827116164732, −11.611102822203058719512601346138, −10.648376180338653331307458655584, −8.78687613335843159966004725346, −7.991646119118360141287556887781, −6.02528795397913188307632731232, −4.92287783380831404318366442017, −3.718156601500054867655695756831, −2.69413922613578286186275935413, 1.69686547510572524689557605466, 3.336051982690506908602513764203, 4.25241234557315480131851336918, 6.292413674731973940788531330714, 7.19402926419863092612230342341, 8.13147567576774478740235879518, 10.329827969296025595795288806714, 11.44370712366581548193255783818, 12.580298724164530453868449325479, 13.4156941467817015553555385392, 14.532299098895774005701995997128, 15.3725614737811332391136044819, 16.74742203204883504621318714585, 18.30305131077991120518812689964, 19.55439587404531241362248133204, 20.038521165885873483603152200488, 21.20428166233389482757785817021, 22.816283113654393971592438402394, 23.59149708940342630162962113115, 23.851878612754449958252817401386, 25.67707256442549897280254718810, 25.97226722430560032317171427011, 27.76128292141747267586481965276, 29.01213702292950176394130462595, 30.104517734637282831697499300433

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