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Properties

Label 1-103-103.73-r1-0-0
Degree $1$
Conductor $103$
Sign $0.563 - 0.826i$
Analytic cond. $11.0688$
Root an. cond. $11.0688$
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  + (0.0922 + 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)  = 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.563 - 0.826i$
Analytic conductor: \(11.0688\)
Root analytic conductor: \(11.0688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (1:\ ),\ 0.563 - 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2963556733 - 0.1566854398i\)
\(L(\frac12)\) \(\approx\) \(0.2963556733 - 0.1566854398i\)
\(L(1)\) \(\approx\) \(0.5445885388 + 0.2318263668i\)
\(L(1)\) \(\approx\) \(0.5445885388 + 0.2318263668i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (0.0922 + 0.995i)T \)
3 \( 1 + (-0.739 - 0.673i)T \)
5 \( 1 + (-0.445 + 0.895i)T \)
7 \( 1 + (-0.273 + 0.961i)T \)
11 \( 1 + (-0.0922 - 0.995i)T \)
13 \( 1 + (-0.273 + 0.961i)T \)
17 \( 1 + (-0.602 - 0.798i)T \)
19 \( 1 + (0.739 - 0.673i)T \)
23 \( 1 + (0.0922 - 0.995i)T \)
29 \( 1 + (0.445 - 0.895i)T \)
31 \( 1 + (-0.932 + 0.361i)T \)
37 \( 1 + (0.850 + 0.526i)T \)
41 \( 1 + (0.445 + 0.895i)T \)
43 \( 1 + (0.850 - 0.526i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.739 + 0.673i)T \)
59 \( 1 + (-0.273 - 0.961i)T \)
61 \( 1 + (-0.602 - 0.798i)T \)
67 \( 1 + (0.273 + 0.961i)T \)
71 \( 1 + (-0.445 - 0.895i)T \)
73 \( 1 + (-0.445 - 0.895i)T \)
79 \( 1 + (0.445 - 0.895i)T \)
83 \( 1 + (-0.273 + 0.961i)T \)
89 \( 1 + (0.982 + 0.183i)T \)
97 \( 1 + (-0.602 + 0.798i)T \)
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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

−29.37731927846507578504088085264, −28.76528648842252988043142465063, −27.64040237383404825752339036062, −27.238608494597770050020653961546, −25.94569020705985022998977395890, −24.05204226577312268799676759882, −23.15643735539858146292941983598, −22.49662202068841882157855964776, −21.218669291332602012794555266337, −20.23533921980316339556287945748, −19.7764851294541016595716255565, −17.887489441694356069132391311841, −17.20645234550799258071883928435, −16.00531889395205456790545159932, −14.740790769717097335228564570139, −13.03272803499928150126010537224, −12.39568702424071599350180477955, −11.150768733249190145816011580319, −10.16813118328798696392720119779, −9.27925963931742135528692111142, −7.64279702998114240667091374789, −5.54846934943920923449934625346, −4.47908301847112569077649810047, −3.588359741022012977079756391764, −1.20651956993615978726494099562, 0.18058394669414535889002680303, 2.79594192239986367776897394936, 4.73865452344779569202803250118, 6.10411758876182510012632888800, 6.79513258424551801700055742784, 7.97780523152664385004357603270, 9.32593199717113903326837574343, 11.121514993317225323657286440320, 12.064318914980667974464792785109, 13.4174131877723327465618667467, 14.410409264474270485275945999290, 15.75734118375755032337412674435, 16.44262200102669722078136626973, 17.899430242422716964119398991584, 18.59982327218795011841998246967, 19.28696545349966279444471575609, 21.82579461748884776495111375730, 22.22560481871386229223404325201, 23.302556263169935408176043049297, 24.25935515437152267390048241606, 24.97984635125931676887554379733, 26.31746982165638226890465478126, 27.080822682479981410862154157950, 28.36014279328314199065432519041, 29.310880017958305449128395022713

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