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Properties

Label 1-101-101.84-r0-0-0
Degree $1$
Conductor $101$
Sign $0.913 + 0.406i$
Analytic cond. $0.469042$
Root an. cond. $0.469042$
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  + (0.309 − 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + 6-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + 10-s + (−0.809 + 0.587i)11-s + (0.309 − 0.951i)12-s + (0.309 − 0.951i)13-s + 14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + 17-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + 6-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + 10-s + (−0.809 + 0.587i)11-s + (0.309 − 0.951i)12-s + (0.309 − 0.951i)13-s + 14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.913 + 0.406i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.913 + 0.406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.170300398 + 0.2484385405i\)
\(L(\frac12)\) \(\approx\) \(1.170300398 + 0.2484385405i\)
\(L(1)\) \(\approx\) \(1.213473056 + 0.05008439967i\)
\(L(1)\) \(\approx\) \(1.213473056 + 0.05008439967i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−29.87620175434498068675722650546, −29.05197035514916823333942618866, −27.58432158030991082025459322339, −26.3542675731647677982998404983, −25.53302175650120354288368258867, −24.50510127369395816930649794236, −23.72043353292465888555584384061, −23.237078440033615418448459083037, −21.34882122430131181024677338210, −20.57389622393236254293260230852, −19.06260599883350819428205565478, −18.011478517743716218000011403262, −16.90852810903864632338590304486, −16.250393458663906533142775676115, −14.48860550409057274476577392708, −13.66192481285689082680881346675, −12.99203695053520790697848827811, −11.73219341950956844047685435940, −9.635556568687012088472503073686, −8.29895243302748370050690090822, −7.642610502334181228000906996659, −6.2576938267698971749764155035, −5.09543531747327189959110991111, −3.5358446117796721455320123600, −1.24579123951389486967156515188, 2.44721625696651235472626437450, 3.13847628243686989161624626094, 4.83151864812607189909974953418, 5.80185667505013925215786195683, 8.0965103601528089956776476951, 9.4407338783349414660916767570, 10.36866350110202421976480001728, 11.16592547469801879943249808355, 12.54690543629199417502796708609, 13.93181913363197132862936562730, 14.91131410399127854159114538800, 15.590738296272601811126617759994, 17.65648610037278988327969632888, 18.456952405189809032225188573116, 19.620951471228520162545972515193, 20.888212914121227815859652069477, 21.4203086683588974765130578912, 22.46842413990106402050849469310, 23.1305562915006858384308198831, 25.00144736241010803147198251479, 26.08057886679188820838379962877, 27.042123944678512326345494488473, 28.07946364333104692089147440093, 28.725556036927406102794550293329, 30.2859359906572880424127981842

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