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Properties

Label 1-103-103.50-r0-0-0
Degree $1$
Conductor $103$
Sign $-0.948 + 0.315i$
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  + (0.552 − 0.833i)2-s + (−0.850 − 0.526i)3-s + (−0.389 − 0.920i)4-s + (−0.952 + 0.303i)5-s + (−0.908 + 0.417i)6-s + (0.332 − 0.943i)7-s + (−0.982 − 0.183i)8-s + (0.445 + 0.895i)9-s + (−0.273 + 0.961i)10-s + (−0.998 − 0.0615i)11-s + (−0.153 + 0.988i)12-s + (−0.982 + 0.183i)13-s + (−0.602 − 0.798i)14-s + (0.969 + 0.243i)15-s + (−0.696 + 0.717i)16-s + (−0.908 − 0.417i)17-s + ⋯
L(s)  = 1  + (0.552 − 0.833i)2-s + (−0.850 − 0.526i)3-s + (−0.389 − 0.920i)4-s + (−0.952 + 0.303i)5-s + (−0.908 + 0.417i)6-s + (0.332 − 0.943i)7-s + (−0.982 − 0.183i)8-s + (0.445 + 0.895i)9-s + (−0.273 + 0.961i)10-s + (−0.998 − 0.0615i)11-s + (−0.153 + 0.988i)12-s + (−0.982 + 0.183i)13-s + (−0.602 − 0.798i)14-s + (0.969 + 0.243i)15-s + (−0.696 + 0.717i)16-s + (−0.908 − 0.417i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $-0.948 + 0.315i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ -0.948 + 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08507812810 - 0.5251244757i\)
\(L(\frac12)\) \(\approx\) \(-0.08507812810 - 0.5251244757i\)
\(L(1)\) \(\approx\) \(0.4645493177 - 0.5430182857i\)
\(L(1)\) \(\approx\) \(0.4645493177 - 0.5430182857i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (0.552 - 0.833i)T \)
3 \( 1 + (-0.850 - 0.526i)T \)
5 \( 1 + (-0.952 + 0.303i)T \)
7 \( 1 + (0.332 - 0.943i)T \)
11 \( 1 + (-0.998 - 0.0615i)T \)
13 \( 1 + (-0.982 + 0.183i)T \)
17 \( 1 + (-0.908 - 0.417i)T \)
19 \( 1 + (0.881 + 0.473i)T \)
23 \( 1 + (0.445 - 0.895i)T \)
29 \( 1 + (0.213 - 0.976i)T \)
31 \( 1 + (-0.273 - 0.961i)T \)
37 \( 1 + (0.932 - 0.361i)T \)
41 \( 1 + (-0.952 - 0.303i)T \)
43 \( 1 + (-0.153 - 0.988i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.0307 - 0.999i)T \)
59 \( 1 + (0.332 + 0.943i)T \)
61 \( 1 + (0.0922 + 0.995i)T \)
67 \( 1 + (0.650 - 0.759i)T \)
71 \( 1 + (0.213 + 0.976i)T \)
73 \( 1 + (0.739 - 0.673i)T \)
79 \( 1 + (0.739 + 0.673i)T \)
83 \( 1 + (0.650 + 0.759i)T \)
89 \( 1 + (-0.602 - 0.798i)T \)
97 \( 1 + (-0.908 + 0.417i)T \)
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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

−30.86372986451147707810692789224, −29.19629542233549468721861093709, −28.161840487473463908507478160680, −27.14331513865616671569807213862, −26.46239333959993854009497125555, −24.90756276800891340643244666706, −23.94935716952449101952183622885, −23.31756199319869603206648004726, −22.08488281941147336945384641923, −21.53372983218546735376178890820, −20.13680621323771904355904817910, −18.37133837519676998518445072993, −17.51569378026769927620997269510, −16.25672076095989376481393446134, −15.50923763739175738045729744839, −14.89646158266677921878310339338, −12.96671694596107660270513227488, −12.10194602798016705978875512616, −11.16456172403989871568464737658, −9.33130206113963199556079046742, −8.07979107615386788158326312544, −6.86443930301995317702622795807, −5.23777809730709445207156494658, −4.833499161841192284003747507088, −3.17980284864303815931234048417, 0.48508699896719026057865383216, 2.44923933374527291725344338351, 4.20224090372079612879989938342, 5.13977181645768240686760614029, 6.81633107481187680283129183576, 7.873225393341688111123112596764, 10.05839919551930052705209722377, 11.01300467588240849722028413992, 11.753786712174194921013192109164, 12.858217748357201674987500576500, 13.88275623320648376501032705839, 15.215033836867520214409571734535, 16.511564076848084831385323161293, 17.92719204563573427049508196148, 18.80676038666306257187556015914, 19.81957431961992643498129917204, 20.806102977195917618984280372625, 22.31157431313003558057087089109, 22.8733180448756914605147125629, 23.88633272861873242401632549667, 24.38420320555338429174127600250, 26.75527679605884609718860816643, 27.20411278298005568527321002150, 28.64847234538053781771830444437, 29.18744362856822601776510450218

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