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Properties

Label 1-4235-4235.109-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.910 + 0.414i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (0.580 + 0.814i)2-s + (0.5 + 0.866i)3-s + (−0.327 + 0.945i)4-s + (−0.415 + 0.909i)6-s + (−0.959 + 0.281i)8-s + (−0.5 + 0.866i)9-s + (−0.981 + 0.189i)12-s + (−0.654 − 0.755i)13-s + (−0.786 − 0.618i)16-s + (0.0475 − 0.998i)17-s + (−0.995 + 0.0950i)18-s + (−0.0475 − 0.998i)19-s + (0.786 + 0.618i)23-s + (−0.723 − 0.690i)24-s + (0.235 − 0.971i)26-s − 27-s + ⋯
L(s)  = 1  + (0.580 + 0.814i)2-s + (0.5 + 0.866i)3-s + (−0.327 + 0.945i)4-s + (−0.415 + 0.909i)6-s + (−0.959 + 0.281i)8-s + (−0.5 + 0.866i)9-s + (−0.981 + 0.189i)12-s + (−0.654 − 0.755i)13-s + (−0.786 − 0.618i)16-s + (0.0475 − 0.998i)17-s + (−0.995 + 0.0950i)18-s + (−0.0475 − 0.998i)19-s + (0.786 + 0.618i)23-s + (−0.723 − 0.690i)24-s + (0.235 − 0.971i)26-s − 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.910 + 0.414i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.910 + 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6453035531 + 2.977403486i\)
\(L(\frac12)\) \(\approx\) \(0.6453035531 + 2.977403486i\)
\(L(1)\) \(\approx\) \(1.047976385 + 1.050235208i\)
\(L(1)\) \(\approx\) \(1.047976385 + 1.050235208i\)

Euler product

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

−18.30642411167104264627483341321, −17.23852053536668566427190357484, −16.84156162435513362774688853955, −15.51909322411941491684822996814, −14.91267445757038464146037039299, −14.34510543879232260385999851124, −13.81417264070222925611691888383, −13.01018128106145791091713636473, −12.53391108517764670375542693864, −11.92858936401640939192172615645, −11.30317197305551362036729211213, −10.372557207948432423517605400100, −9.75216584761611588406534142556, −8.90669592490652556211743400922, −8.35857195258715918701719746567, −7.36033930156512252283068609015, −6.65110556658756172595524681221, −5.93171675026922193493651498348, −5.21825008286730115422530033674, −4.038839767219266260285950364109, −3.736939522794516556832920530950, −2.53694305363929543250243823849, −2.16030915821306659756514509285, −1.307620450274298615473482996477, −0.45937385046367585292611517162, 0.65451913000757516327459491001, 2.31397194521109767574510011361, 3.00976693816712193359902896879, 3.507580002477281908908162022863, 4.605846956073646093467473583768, 5.013845863969259051476651334775, 5.5646846008791065076721172045, 6.7334913021642215378293752373, 7.29293163036263752854332131532, 8.1205233099460048911144427332, 8.67489211973416650237413523532, 9.547745900924490871473502898786, 9.93314142362535824272800464306, 11.15365259048084434521172849795, 11.57380274626830097849629516209, 12.61689940563293910283963761082, 13.34055248138195776382688676220, 13.79608925218150625236797646472, 14.62820906868978609834391827475, 15.21450344478940340013142048295, 15.536291227544295213093653947098, 16.36502523341991402546482795797, 16.95208314354376564123141596426, 17.5428030104676312127891428194, 18.3402768083517686994018640299

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