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Properties

Label 1-103-103.2-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

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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: $\chi_{103} (2, \cdot )$
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 \( 1 + (0.332 - 0.943i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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