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Properties

Label 1-4235-4235.2168-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.667 - 0.744i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.814 − 0.580i)2-s + (0.866 + 0.5i)3-s + (0.327 + 0.945i)4-s + (−0.415 − 0.909i)6-s + (0.281 − 0.959i)8-s + (0.5 + 0.866i)9-s + (−0.189 + 0.981i)12-s + (0.755 + 0.654i)13-s + (−0.786 + 0.618i)16-s + (−0.998 + 0.0475i)17-s + (0.0950 − 0.995i)18-s + (0.0475 − 0.998i)19-s + (−0.618 − 0.786i)23-s + (0.723 − 0.690i)24-s + (−0.235 − 0.971i)26-s + i·27-s + ⋯
L(s)  = 1  + (−0.814 − 0.580i)2-s + (0.866 + 0.5i)3-s + (0.327 + 0.945i)4-s + (−0.415 − 0.909i)6-s + (0.281 − 0.959i)8-s + (0.5 + 0.866i)9-s + (−0.189 + 0.981i)12-s + (0.755 + 0.654i)13-s + (−0.786 + 0.618i)16-s + (−0.998 + 0.0475i)17-s + (0.0950 − 0.995i)18-s + (0.0475 − 0.998i)19-s + (−0.618 − 0.786i)23-s + (0.723 − 0.690i)24-s + (−0.235 − 0.971i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.667 - 0.744i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.667 - 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.301126923 - 0.5813858400i\)
\(L(\frac12)\) \(\approx\) \(1.301126923 - 0.5813858400i\)
\(L(1)\) \(\approx\) \(0.9603667112 - 0.1021039199i\)
\(L(1)\) \(\approx\) \(0.9603667112 - 0.1021039199i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.814 - 0.580i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.755 + 0.654i)T \)
17 \( 1 + (-0.998 + 0.0475i)T \)
19 \( 1 + (0.0475 - 0.998i)T \)
23 \( 1 + (-0.618 - 0.786i)T \)
29 \( 1 + (-0.841 - 0.540i)T \)
31 \( 1 + (-0.981 + 0.189i)T \)
37 \( 1 + (0.945 + 0.327i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (-0.0950 - 0.995i)T \)
53 \( 1 + (0.618 - 0.786i)T \)
59 \( 1 + (0.580 + 0.814i)T \)
61 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (-0.0950 + 0.995i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (0.371 - 0.928i)T \)
79 \( 1 + (-0.723 - 0.690i)T \)
83 \( 1 + (0.989 + 0.142i)T \)
89 \( 1 + (-0.888 - 0.458i)T \)
97 \( 1 + (0.281 - 0.959i)T \)
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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

−18.25068782142421859499749780671, −18.09278261964003189294534317184, −17.25125789226024841797167116243, −16.3139430806679421732875004200, −15.82604020821233766521110195055, −15.04408137664001032980101594908, −14.5942901758456281551801783041, −13.82355567929787902582105319709, −13.16682965182174209434478532717, −12.50839050381980770786597402296, −11.38225095661515057284502867482, −10.89686476038929011312008283626, −9.87327195023173385758214163136, −9.39761979047628400715756987442, −8.69096657780990580564986340349, −7.94289320294698947507399445770, −7.64653851691381322834859494508, −6.69196894236425800608257575569, −6.09060674765150538980149134300, −5.37096168004560219092858889208, −4.17003305347399007566898673631, −3.408789667947780521284055832537, −2.38721763821532125180055099680, −1.67924111492641370814870308155, −0.92085093479442707521175370986, 0.517856312459372583303884002107, 1.920059995654358429504603430123, 2.18185187227886117800032368073, 3.16485527032258371613993369496, 4.004437767287058334985952474477, 4.36325833277042828246411637837, 5.58667711103155973390974030757, 6.824577638929559700372436909958, 7.17973202794147692219013220446, 8.29999077093401394622616560292, 8.66398765839540528716018715234, 9.2538398240729792353038413990, 9.9280341973063335102118323170, 10.70546141113353967367184162281, 11.214732702242529326886480334054, 11.92492280348504459867965758430, 13.064381761691957627257827772827, 13.31889091038946993376589136544, 14.15021000510083730767188387753, 15.09270110157912609248394309242, 15.64416422499486331960343073800, 16.344873392937305658907552943597, 16.83405826101128523837058786951, 17.82613937584504752216767677452, 18.37140719794193510189039276559

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