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

Degree: \(1\)
Conductor: \(103\)
Sign: $0.534 + 0.845i$
Motivic weight: \(0\)
Character: $\chi_{103} (52, \cdot )$
Sato-Tate group: $\mu(51)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.534 + 0.845i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.553493494 + 0.8554101287i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.553493494 + 0.8554101287i\)
\(L(\chi,1)\) \(\approx\) \(1.624580755 + 0.5850154404i\)
\(L(1,\chi)\) \(\approx\) \(1.624580755 + 0.5850154404i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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