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Properties

Label 1-101-101.4-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: $\chi_{101} (4, \cdot )$
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

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

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