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Properties

Label 1-4235-4235.1418-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.990 - 0.136i$
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  + (0.971 + 0.235i)2-s + (0.866 − 0.5i)3-s + (0.888 + 0.458i)4-s + (0.959 − 0.281i)6-s + (0.755 + 0.654i)8-s + (0.5 − 0.866i)9-s + (0.998 − 0.0475i)12-s + (−0.540 − 0.841i)13-s + (0.580 + 0.814i)16-s + (0.371 + 0.928i)17-s + (0.690 − 0.723i)18-s + (0.928 + 0.371i)19-s + (−0.814 + 0.580i)23-s + (0.981 + 0.189i)24-s + (−0.327 − 0.945i)26-s − i·27-s + ⋯
L(s)  = 1  + (0.971 + 0.235i)2-s + (0.866 − 0.5i)3-s + (0.888 + 0.458i)4-s + (0.959 − 0.281i)6-s + (0.755 + 0.654i)8-s + (0.5 − 0.866i)9-s + (0.998 − 0.0475i)12-s + (−0.540 − 0.841i)13-s + (0.580 + 0.814i)16-s + (0.371 + 0.928i)17-s + (0.690 − 0.723i)18-s + (0.928 + 0.371i)19-s + (−0.814 + 0.580i)23-s + (0.981 + 0.189i)24-s + (−0.327 − 0.945i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.990 - 0.136i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.990 - 0.136i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.235007675 - 0.3581696062i\)
\(L(\frac12)\) \(\approx\) \(5.235007675 - 0.3581696062i\)
\(L(1)\) \(\approx\) \(2.683322254 - 0.04695072996i\)
\(L(1)\) \(\approx\) \(2.683322254 - 0.04695072996i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.971 + 0.235i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.540 - 0.841i)T \)
17 \( 1 + (0.371 + 0.928i)T \)
19 \( 1 + (0.928 + 0.371i)T \)
23 \( 1 + (-0.814 + 0.580i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (0.0475 - 0.998i)T \)
37 \( 1 + (0.458 + 0.888i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 + (-0.690 - 0.723i)T \)
53 \( 1 + (0.814 + 0.580i)T \)
59 \( 1 + (-0.235 - 0.971i)T \)
61 \( 1 + (-0.723 + 0.690i)T \)
67 \( 1 + (0.690 - 0.723i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.0950 + 0.995i)T \)
79 \( 1 + (0.981 - 0.189i)T \)
83 \( 1 + (0.909 - 0.415i)T \)
89 \( 1 + (0.786 + 0.618i)T \)
97 \( 1 + (0.755 + 0.654i)T \)
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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

−18.59231211417994691171579158894, −17.83610511897489910562567579616, −16.46455295734437172828455279240, −16.2704302292152941698055599638, −15.64209725955731402388256239456, −14.72991210420947526406354009297, −14.17350251343100920121035286652, −13.97935412760070081992226308837, −13.02933610823045137603119220812, −12.33717665731357539240874527770, −11.66155603166835623965091426042, −10.88513518382773865001866558188, −10.186325646253482963077134964021, −9.459283332653441059324324949945, −8.93855168513134773012003005588, −7.71673251845574781559041544817, −7.29996042952252062047021968600, −6.48780901498611435503955007851, −5.39913688304929725500837561654, −4.86121826318079020008711487044, −4.16381720012424216098547157233, −3.42089806170653224099781227232, −2.68706764278324965274604455538, −2.102291704141761549608400584444, −1.08394147300706458008622997979, 1.00415872996685942656750464593, 1.98498312183229145214483154867, 2.619384504969419501696563186697, 3.46542658679416983797847699475, 3.954733267397739294018907681786, 4.909181601586735330849962241874, 5.901300004628672161740316986639, 6.23798297276587397438111408440, 7.408327556953300202789229439346, 7.79746608044578163229240245922, 8.24170769667725276435248004468, 9.47607481383467547750005675672, 10.020949551552541506398239487072, 10.965487739626501512884915761270, 11.98267279782560200735847828080, 12.251771653255776409580045229445, 13.18828696369818264071109453349, 13.49592813248171564324114804322, 14.31765467909039063565100603102, 14.88684635780891871019720739475, 15.374169669720438137836476769967, 16.07832340801421355785912649073, 16.98617777409035317710977106836, 17.60760985531645421694523486935, 18.397535230560360610855816400111

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