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Properties

Label 1-101-101.74-r1-0-0
Degree $1$
Conductor $101$
Sign $0.804 - 0.594i$
Analytic cond. $10.8539$
Root an. cond. $10.8539$
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  + (−0.904 − 0.425i)2-s + (−0.481 + 0.876i)3-s + (0.637 + 0.770i)4-s + (−0.425 − 0.904i)5-s + (0.809 − 0.587i)6-s + (0.684 + 0.728i)7-s + (−0.248 − 0.968i)8-s + (−0.535 − 0.844i)9-s + i·10-s + (−0.844 + 0.535i)11-s + (−0.982 + 0.187i)12-s + (−0.728 − 0.684i)13-s + (−0.309 − 0.951i)14-s + (0.998 + 0.0627i)15-s + (−0.187 + 0.982i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.904 − 0.425i)2-s + (−0.481 + 0.876i)3-s + (0.637 + 0.770i)4-s + (−0.425 − 0.904i)5-s + (0.809 − 0.587i)6-s + (0.684 + 0.728i)7-s + (−0.248 − 0.968i)8-s + (−0.535 − 0.844i)9-s + i·10-s + (−0.844 + 0.535i)11-s + (−0.982 + 0.187i)12-s + (−0.728 − 0.684i)13-s + (−0.309 − 0.951i)14-s + (0.998 + 0.0627i)15-s + (−0.187 + 0.982i)16-s + (0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.804 - 0.594i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ 0.804 - 0.594i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7132556451 - 0.2350991263i\)
\(L(\frac12)\) \(\approx\) \(0.7132556451 - 0.2350991263i\)
\(L(1)\) \(\approx\) \(0.6051019680 - 0.04076710260i\)
\(L(1)\) \(\approx\) \(0.6051019680 - 0.04076710260i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.904 - 0.425i)T \)
3 \( 1 + (-0.481 + 0.876i)T \)
5 \( 1 + (-0.425 - 0.904i)T \)
7 \( 1 + (0.684 + 0.728i)T \)
11 \( 1 + (-0.844 + 0.535i)T \)
13 \( 1 + (-0.728 - 0.684i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.187 - 0.982i)T \)
23 \( 1 + (0.992 + 0.125i)T \)
29 \( 1 + (0.684 - 0.728i)T \)
31 \( 1 + (0.728 - 0.684i)T \)
37 \( 1 + (0.876 - 0.481i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (0.929 - 0.368i)T \)
47 \( 1 + (0.929 + 0.368i)T \)
53 \( 1 + (0.770 + 0.637i)T \)
59 \( 1 + (-0.982 - 0.187i)T \)
61 \( 1 + (0.770 - 0.637i)T \)
67 \( 1 + (0.481 + 0.876i)T \)
71 \( 1 + (0.876 + 0.481i)T \)
73 \( 1 + (0.125 - 0.992i)T \)
79 \( 1 + (-0.992 + 0.125i)T \)
83 \( 1 + (-0.125 - 0.992i)T \)
89 \( 1 + (0.982 - 0.187i)T \)
97 \( 1 + (-0.637 - 0.770i)T \)
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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

−29.498934115655881706464725273206, −28.907567696551576774624436015529, −27.33277762425241971698934697475, −26.87092641220910005132939383346, −25.66100928727674704466806731560, −24.55900651703327075412144180734, −23.55140803962385573922601779575, −23.07318003155586074732617434203, −21.26117513663340747847253770799, −19.81443801725579593739193626871, −18.81768351951456655398657336095, −18.27719917879702231964423652298, −17.11319778005685615708418741231, −16.252463049163275167086847950845, −14.673242282736782270964704405449, −13.90157500223168628232933398215, −11.99961923940502239741413698592, −11.037247739182040561177525162628, −10.21175768750950884077428567892, −8.20584880070181210034873711891, −7.45986213917734982697203058353, −6.591929670892994274015453693615, −5.125886034182924176597035072298, −2.64679563803332768815309958478, −0.99478015079236988631393700877, 0.61908326583829627961351954808, 2.64373553364638909477198774383, 4.42946447666619805952964010904, 5.54000655796477571508659344102, 7.647105442368929119095411290906, 8.677304944338431855065017177231, 9.72073580071715135184090346638, 10.847203967402151798764578871586, 11.93914567479083133383118185986, 12.716911119469544069120202631549, 15.197795294611448834031130582478, 15.65987203468873407818120734762, 17.08404215323259628340159082399, 17.56644843061512050627696538581, 19.01048701515075402885318612295, 20.2739544489172979549843730622, 21.01036789382216202901903437046, 21.79098654957761239499067080005, 23.28904345458530695535793613449, 24.51679843485006819709438187301, 25.62283172866082883749596758602, 26.85373276963634835730666232336, 27.671406182788779698674165105300, 28.275151107915550189592089845556, 28.99824832129461578117074379449

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