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Properties

Label 1-101-101.95-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: $\chi_{101} (95, \cdot )$
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

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

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