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Properties

Label 1-103-103.24-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: $\chi_{103} (24, \cdot )$
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

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

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