↑

Properties

Label 1-103-103.52-r0-0-0
Degree $1$
Conductor $103$
Sign $0.534 + 0.845i$
Analytic cond. $0.478329$
Root an. cond. $0.478329$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.534 + 0.845i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.534 + 0.845i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (0.992 + 0.122i)T \)
3 \( 1 + (0.445 + 0.895i)T \)
5 \( 1 + (-0.908 - 0.417i)T \)
7 \( 1 + (-0.153 + 0.988i)T \)
11 \( 1 + (-0.389 - 0.920i)T \)
13 \( 1 + (0.932 - 0.361i)T \)
17 \( 1 + (0.332 - 0.943i)T \)
19 \( 1 + (-0.998 + 0.0615i)T \)
23 \( 1 + (-0.602 - 0.798i)T \)
29 \( 1 + (0.816 - 0.577i)T \)
31 \( 1 + (-0.850 + 0.526i)T \)
37 \( 1 + (0.739 - 0.673i)T \)
41 \( 1 + (-0.908 + 0.417i)T \)
43 \( 1 + (0.213 - 0.976i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.552 + 0.833i)T \)
59 \( 1 + (-0.153 - 0.988i)T \)
61 \( 1 + (-0.982 + 0.183i)T \)
67 \( 1 + (-0.779 + 0.626i)T \)
71 \( 1 + (0.816 + 0.577i)T \)
73 \( 1 + (0.0922 - 0.995i)T \)
79 \( 1 + (0.0922 + 0.995i)T \)
83 \( 1 + (-0.779 - 0.626i)T \)
89 \( 1 + (-0.273 + 0.961i)T \)
97 \( 1 + (0.332 + 0.943i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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