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Properties

Label 1-101-101.93-r1-0-0
Degree $1$
Conductor $101$
Sign $-0.941 - 0.337i$
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  + (−0.982 − 0.187i)2-s + (−0.904 − 0.425i)3-s + (0.929 + 0.368i)4-s + (−0.187 − 0.982i)5-s + (0.809 + 0.587i)6-s + (0.125 − 0.992i)7-s + (−0.844 − 0.535i)8-s + (0.637 + 0.770i)9-s + i·10-s + (0.770 − 0.637i)11-s + (−0.684 − 0.728i)12-s + (0.992 − 0.125i)13-s + (−0.309 + 0.951i)14-s + (−0.248 + 0.968i)15-s + (0.728 + 0.684i)16-s + (0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (−0.982 − 0.187i)2-s + (−0.904 − 0.425i)3-s + (0.929 + 0.368i)4-s + (−0.187 − 0.982i)5-s + (0.809 + 0.587i)6-s + (0.125 − 0.992i)7-s + (−0.844 − 0.535i)8-s + (0.637 + 0.770i)9-s + i·10-s + (0.770 − 0.637i)11-s + (−0.684 − 0.728i)12-s + (0.992 − 0.125i)13-s + (−0.309 + 0.951i)14-s + (−0.248 + 0.968i)15-s + (0.728 + 0.684i)16-s + (0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.941 - 0.337i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ -0.941 - 0.337i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1252119593 - 0.7209079804i\)
\(L(\frac12)\) \(\approx\) \(0.1252119593 - 0.7209079804i\)
\(L(1)\) \(\approx\) \(0.4729930304 - 0.3554763936i\)
\(L(1)\) \(\approx\) \(0.4729930304 - 0.3554763936i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (-0.982 - 0.187i)T \)
3 \( 1 + (-0.904 - 0.425i)T \)
5 \( 1 + (-0.187 - 0.982i)T \)
7 \( 1 + (0.125 - 0.992i)T \)
11 \( 1 + (0.770 - 0.637i)T \)
13 \( 1 + (0.992 - 0.125i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.728 - 0.684i)T \)
23 \( 1 + (-0.876 - 0.481i)T \)
29 \( 1 + (0.125 + 0.992i)T \)
31 \( 1 + (-0.992 - 0.125i)T \)
37 \( 1 + (-0.425 - 0.904i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (-0.0627 + 0.998i)T \)
47 \( 1 + (-0.0627 - 0.998i)T \)
53 \( 1 + (0.368 + 0.929i)T \)
59 \( 1 + (-0.684 + 0.728i)T \)
61 \( 1 + (0.368 - 0.929i)T \)
67 \( 1 + (0.904 - 0.425i)T \)
71 \( 1 + (-0.425 + 0.904i)T \)
73 \( 1 + (-0.481 + 0.876i)T \)
79 \( 1 + (0.876 - 0.481i)T \)
83 \( 1 + (0.481 + 0.876i)T \)
89 \( 1 + (0.684 + 0.728i)T \)
97 \( 1 + (-0.929 - 0.368i)T \)
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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

−29.88773350289455021637518499688, −28.7272554192674939902620469267, −27.879083927375173936996750811644, −27.29288480187477957181510659247, −26.05542804017471976435171315234, −25.25107618563321952818179918973, −23.87521574377781752593815846566, −22.81002969264174427787964006606, −21.792941248563544167200538316332, −20.69225968645783706322166527637, −19.14276433226020458387789145627, −18.30943013512889005523292889427, −17.59169198412759923804733058565, −16.29469815844586319603743685613, −15.40771083074179582643095370543, −14.53415015205372094039414328500, −12.083842093136669177112455967958, −11.4893515072725365708136741256, −10.30185235930578220786928187521, −9.40197629567495547955185638806, −7.858927287705547134578521446778, −6.46586510334847381726650913293, −5.72584128348305858375228643367, −3.58073643819869342066506904411, −1.63741330143079935155928702782, 0.577510190587569467842655238345, 1.34420790701120858288573310584, 3.82907592728410702492876090734, 5.5829502335369038192606417163, 6.94523943453239914499352224023, 8.00828280640849472768298333186, 9.27886697728035020246422299005, 10.680236193878542459409797406426, 11.54534542530792008993094365608, 12.56177435189002409434463604272, 13.83573288677692979468567460053, 16.15126371553559513000365859250, 16.428928393550015947908597591377, 17.4888988386021637500355614656, 18.42296896128008323071448890602, 19.69236421802572952974541934008, 20.47490324287405453355553039016, 21.69711201483832972291673880214, 23.19905252918668120255682901768, 24.15739637281576043092814420900, 24.89587230104861013934740362511, 26.28601596359941543315639622435, 27.60205713796196683372396996744, 27.91158179500030783221992722978, 29.154292879489599379315846374602

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