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Properties

Label 1-101-101.19-r0-0-0
Degree $1$
Conductor $101$
Sign $0.955 + 0.295i$
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  + (0.968 − 0.248i)2-s + (0.0627 + 0.998i)3-s + (0.876 − 0.481i)4-s + (0.968 + 0.248i)5-s + (0.309 + 0.951i)6-s + (−0.637 − 0.770i)7-s + (0.728 − 0.684i)8-s + (−0.992 + 0.125i)9-s + 10-s + (−0.992 + 0.125i)11-s + (0.535 + 0.844i)12-s + (−0.637 + 0.770i)13-s + (−0.809 − 0.587i)14-s + (−0.187 + 0.982i)15-s + (0.535 − 0.844i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.968 − 0.248i)2-s + (0.0627 + 0.998i)3-s + (0.876 − 0.481i)4-s + (0.968 + 0.248i)5-s + (0.309 + 0.951i)6-s + (−0.637 − 0.770i)7-s + (0.728 − 0.684i)8-s + (−0.992 + 0.125i)9-s + 10-s + (−0.992 + 0.125i)11-s + (0.535 + 0.844i)12-s + (−0.637 + 0.770i)13-s + (−0.809 − 0.587i)14-s + (−0.187 + 0.982i)15-s + (0.535 − 0.844i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.955 + 0.295i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.955 + 0.295i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.788225442 + 0.2700108739i\)
\(L(\frac12)\) \(\approx\) \(1.788225442 + 0.2700108739i\)
\(L(1)\) \(\approx\) \(1.748518335 + 0.1729121303i\)
\(L(1)\) \(\approx\) \(1.748518335 + 0.1729121303i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.968 - 0.248i)T \)
3 \( 1 + (0.0627 + 0.998i)T \)
5 \( 1 + (0.968 + 0.248i)T \)
7 \( 1 + (-0.637 - 0.770i)T \)
11 \( 1 + (-0.992 + 0.125i)T \)
13 \( 1 + (-0.637 + 0.770i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.535 + 0.844i)T \)
23 \( 1 + (-0.929 - 0.368i)T \)
29 \( 1 + (-0.637 + 0.770i)T \)
31 \( 1 + (-0.637 - 0.770i)T \)
37 \( 1 + (0.0627 - 0.998i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.425 + 0.904i)T \)
47 \( 1 + (-0.425 - 0.904i)T \)
53 \( 1 + (0.876 + 0.481i)T \)
59 \( 1 + (0.535 - 0.844i)T \)
61 \( 1 + (0.876 - 0.481i)T \)
67 \( 1 + (0.0627 - 0.998i)T \)
71 \( 1 + (0.0627 + 0.998i)T \)
73 \( 1 + (-0.929 - 0.368i)T \)
79 \( 1 + (-0.929 + 0.368i)T \)
83 \( 1 + (-0.929 + 0.368i)T \)
89 \( 1 + (0.535 + 0.844i)T \)
97 \( 1 + (0.876 - 0.481i)T \)
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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

−29.914127652386695901493179226636, −29.04721665800216717349117978357, −28.427331172771600269492105700368, −25.98038742839306928698481918287, −25.58459063786378949146355950343, −24.50377160498285524618944299307, −23.86392847692080232315187257880, −22.50878084208742402690861030827, −21.762083837993446771577558908578, −20.54511772378223487968251178946, −19.43883135713825408485053630760, −18.08584344413818325824562237473, −17.14910189418512475877555515572, −15.7953463123951095051553170098, −14.62125266615157815264744458121, −13.339980790887132155530855537256, −12.90265475813589759379995274344, −11.88451234423379772602193928412, −10.22060561587781404163412328696, −8.518583680348902603538744846101, −7.25884785374190314086689450241, −5.91302041745969930838463820698, −5.38906210081989687917340359324, −3.02143465516942877169759598896, −2.0758365413554016433001302784, 2.34658263980680036581361135563, 3.55521427128723170741117455645, 4.894968165750114030361977800011, 5.91653065850971809090894594355, 7.33291878978830114993146622838, 9.675766342191237390781742976856, 10.1343875122478440026963978974, 11.32428344726412811183353748652, 12.85942659126161959413786964709, 13.97614406291640457167513626667, 14.60938159057242896855011652971, 16.136194080109442981653259466294, 16.66326500918510472892736631423, 18.421590957518794401994144041743, 19.99038331545481173489870325363, 20.73849115936339813615594870346, 21.64569751002038986412348771289, 22.49053900213723546783274543512, 23.3404046363435764862079544925, 24.762254214829698977091740975619, 25.92718034535149727341370778504, 26.566012587475671593697916475127, 28.221260659939509906247902189828, 29.179145021052833093718270519317, 29.71546370104593584498342210919

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