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Properties

Label 1-101-101.76-r0-0-0
Degree $1$
Conductor $101$
Sign $0.0787 - 0.996i$
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  + (0.992 − 0.125i)2-s + (−0.728 − 0.684i)3-s + (0.968 − 0.248i)4-s + (−0.992 − 0.125i)5-s + (−0.809 − 0.587i)6-s + (0.425 − 0.904i)7-s + (0.929 − 0.368i)8-s + (0.0627 + 0.998i)9-s − 10-s + (−0.0627 − 0.998i)11-s + (−0.876 − 0.481i)12-s + (−0.425 − 0.904i)13-s + (0.309 − 0.951i)14-s + (0.637 + 0.770i)15-s + (0.876 − 0.481i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.992 − 0.125i)2-s + (−0.728 − 0.684i)3-s + (0.968 − 0.248i)4-s + (−0.992 − 0.125i)5-s + (−0.809 − 0.587i)6-s + (0.425 − 0.904i)7-s + (0.929 − 0.368i)8-s + (0.0627 + 0.998i)9-s − 10-s + (−0.0627 − 0.998i)11-s + (−0.876 − 0.481i)12-s + (−0.425 − 0.904i)13-s + (0.309 − 0.951i)14-s + (0.637 + 0.770i)15-s + (0.876 − 0.481i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0787 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0787 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.0787 - 0.996i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.0787 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9462196432 - 0.8744615541i\)
\(L(\frac12)\) \(\approx\) \(0.9462196432 - 0.8744615541i\)
\(L(1)\) \(\approx\) \(1.178602459 - 0.5777935898i\)
\(L(1)\) \(\approx\) \(1.178602459 - 0.5777935898i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.992 - 0.125i)T \)
3 \( 1 + (-0.728 - 0.684i)T \)
5 \( 1 + (-0.992 - 0.125i)T \)
7 \( 1 + (0.425 - 0.904i)T \)
11 \( 1 + (-0.0627 - 0.998i)T \)
13 \( 1 + (-0.425 - 0.904i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.876 + 0.481i)T \)
23 \( 1 + (-0.187 + 0.982i)T \)
29 \( 1 + (0.425 + 0.904i)T \)
31 \( 1 + (-0.425 + 0.904i)T \)
37 \( 1 + (0.728 - 0.684i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (0.535 + 0.844i)T \)
47 \( 1 + (0.535 - 0.844i)T \)
53 \( 1 + (-0.968 - 0.248i)T \)
59 \( 1 + (-0.876 + 0.481i)T \)
61 \( 1 + (-0.968 + 0.248i)T \)
67 \( 1 + (-0.728 + 0.684i)T \)
71 \( 1 + (0.728 + 0.684i)T \)
73 \( 1 + (0.187 - 0.982i)T \)
79 \( 1 + (-0.187 - 0.982i)T \)
83 \( 1 + (0.187 + 0.982i)T \)
89 \( 1 + (-0.876 - 0.481i)T \)
97 \( 1 + (0.968 - 0.248i)T \)
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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

−30.56347801803573336839299859664, −28.897804887331039426241904052714, −28.34067484682374687264524251506, −27.10037163395251731585328522118, −26.066229152009029814216689601992, −24.51532731978857587049915261236, −23.817788871708991886381210295333, −22.59249327183176805779494410714, −22.19422090624546497026973778038, −20.92880964198564245529242324356, −20.05856831799516244616876743519, −18.482131649948210271057162738647, −17.10778266488734949027901647344, −15.81928856515238297527169705343, −15.36837733391660872714233527599, −14.332114472658037735220388285143, −12.444521155745463827835573161914, −11.789019332168851508626961379853, −11.00125105751516445076521994273, −9.31907171945192753866753816368, −7.54739228192869586840836625247, −6.35719866553689741723646170074, −4.84897090462682898307553158603, −4.296198047672393579711683422397, −2.545944335409677087039323690553, 1.17925291135271363261814116813, 3.27958681910591795925991112121, 4.62478645634993201721978268049, 5.79608695061583103873302262198, 7.21862884536761528715108430885, 7.962800524226840916900035628010, 10.657841116352552658062922736552, 11.25058645824607204761161997796, 12.348208226842308508160369716182, 13.29984380793226252755001222783, 14.389234293402531330384194221968, 15.826282304486835079943469583459, 16.64351578865985135879642391431, 17.97350416281346185453739688447, 19.545232972605748985270381664787, 19.99562697010703194196880714905, 21.56310159255028954052858641581, 22.61860067394970183293247955208, 23.49232872368892738874058276793, 24.08571084005846200590973127812, 24.92945416090164465843820962290, 26.73593420243783098413735127483, 27.74564290558563416399246036372, 29.01767294343384386407514949024, 29.789174985533761438117397422923

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