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Properties

Label 1-101-101.17-r0-0-0
Degree $1$
Conductor $101$
Sign $0.971 - 0.238i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $0.971 - 0.238i$
Analytic conductor: \(0.469042\)
Root analytic conductor: \(0.469042\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (0:\ ),\ 0.971 - 0.238i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7468296308 - 0.09036858034i\)
\(L(\frac12)\) \(\approx\) \(0.7468296308 - 0.09036858034i\)
\(L(1)\) \(\approx\) \(0.8047925708 + 0.03569485903i\)
\(L(1)\) \(\approx\) \(0.8047925708 + 0.03569485903i\)

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

−29.56996374993268980836645273001, −28.62840297973222488712748832915, −28.03147350862790478197960786596, −27.33886249715973232828223975964, −25.959390814515022544235848730375, −25.08101037850738117924140830040, −23.31746899865876629889474112043, −22.29296381990089710736778637088, −21.332446910767120728171061427108, −20.77144905497514324278905625505, −19.60708532078201906493830774767, −18.405762527147245105062429225988, −17.03918129635035734494973721019, −16.500064317885573048401807983408, −14.98842437986667678548744473791, −13.56273618300866521403476394999, −12.04431590849405096424324849703, −11.71815496434747537304954656286, −9.79556818265697487517228630341, −9.47387579173038276228063848068, −8.33208538643289838601975921397, −5.8786682165995717027796526439, −4.682022361470886238993058868548, −3.5137497648522238391948239552, −1.67110624294060194863504570864, 1.02119864862703612205533829186, 3.32944220235167526167075927682, 5.43594286541720158720130629309, 6.611025888793170472653097692344, 7.19431759671925015681290938620, 8.47503282388448124414395796050, 10.11615579549741481524112171712, 11.100341174286706215229766601099, 12.92452243899447693436375797619, 13.90581842888534882483981400676, 14.625733856319034112681964613640, 16.27616179478071977646095590626, 17.23480451391573334753936660046, 18.06323212342240618885808451456, 19.01080434186734276740587275345, 19.923484638619696836403066792587, 22.0844105184204277089608290094, 22.88474625613209813285956728857, 23.62643796780574877014969074212, 24.82347854835811015743263140916, 25.59543728240959264316350398359, 26.563224374865743787626540626162, 27.58041643419809118215143861125, 28.90810361420430155622501698476, 30.00651105516995247184191697140

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