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Properties

Label 1-4235-4235.397-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.374 + 0.927i$
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  + (−0.690 + 0.723i)2-s + (−0.866 − 0.5i)3-s + (−0.0475 − 0.998i)4-s + (0.959 − 0.281i)6-s + (0.755 + 0.654i)8-s + (0.5 + 0.866i)9-s + (−0.458 + 0.888i)12-s + (0.540 + 0.841i)13-s + (−0.995 + 0.0950i)16-s + (−0.618 + 0.786i)17-s + (−0.971 − 0.235i)18-s + (−0.786 + 0.618i)19-s + (0.0950 + 0.995i)23-s + (−0.327 − 0.945i)24-s + (−0.981 − 0.189i)26-s − i·27-s + ⋯
L(s)  = 1  + (−0.690 + 0.723i)2-s + (−0.866 − 0.5i)3-s + (−0.0475 − 0.998i)4-s + (0.959 − 0.281i)6-s + (0.755 + 0.654i)8-s + (0.5 + 0.866i)9-s + (−0.458 + 0.888i)12-s + (0.540 + 0.841i)13-s + (−0.995 + 0.0950i)16-s + (−0.618 + 0.786i)17-s + (−0.971 − 0.235i)18-s + (−0.786 + 0.618i)19-s + (0.0950 + 0.995i)23-s + (−0.327 − 0.945i)24-s + (−0.981 − 0.189i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.374 + 0.927i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.374 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4384772819 + 0.6497741498i\)
\(L(\frac12)\) \(\approx\) \(0.4384772819 + 0.6497741498i\)
\(L(1)\) \(\approx\) \(0.5704346587 + 0.2086173703i\)
\(L(1)\) \(\approx\) \(0.5704346587 + 0.2086173703i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.690 + 0.723i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.540 + 0.841i)T \)
17 \( 1 + (-0.618 + 0.786i)T \)
19 \( 1 + (-0.786 + 0.618i)T \)
23 \( 1 + (0.0950 + 0.995i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (0.888 - 0.458i)T \)
37 \( 1 + (0.998 + 0.0475i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 + (0.971 - 0.235i)T \)
53 \( 1 + (-0.0950 + 0.995i)T \)
59 \( 1 + (0.723 - 0.690i)T \)
61 \( 1 + (-0.235 - 0.971i)T \)
67 \( 1 + (0.971 + 0.235i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (-0.814 + 0.580i)T \)
79 \( 1 + (0.327 - 0.945i)T \)
83 \( 1 + (-0.909 + 0.415i)T \)
89 \( 1 + (0.928 - 0.371i)T \)
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

−17.949899391321773055699114401647, −17.671370578499106852352751475449, −16.981931890678872362364432502409, −16.23147223971389641556870743363, −15.710693929310430413367824118623, −15.06476662846749811292069923312, −13.929221443904358439796168829784, −13.032941343675887967195804749121, −12.64828416633180955842758397160, −11.72241771694523638586173815888, −11.27361600549044738122863195675, −10.56372280270185573748824779439, −10.15007379250869354370102262587, −9.26826491038571113556655708438, −8.707034196534513397195821334061, −7.893141343318622418139714435247, −6.960050941239114704977872086387, −6.34320537070137916828787513399, −5.43840449716725872680368057810, −4.37976577185430111220391626415, −4.14980818569460635954583462860, −2.90462684677262480140348130014, −2.38288648142537607390818115978, −0.995984692823988494907505430356, −0.46353591456156134294236355288, 0.916095138126456104627036143156, 1.6320562598414651725182635622, 2.3762854454457265762329094422, 3.9893200165091355859129887902, 4.58217972589218555878621951419, 5.545498746539663499506409640933, 6.1997629246338704057239468421, 6.56752263417633567744609603634, 7.45568634617844386609633610949, 8.0381563909847089030310600188, 8.8408588540260193406862299912, 9.54090530863257571238100146481, 10.44802587525226828586798787980, 10.970158251277600007825947339220, 11.531221240810511620964029544443, 12.47418313620403870837000899211, 13.19011906071808337859819439437, 13.88382328861559684127961880871, 14.6039849437309668212896332535, 15.45539590969904879387649892138, 16.06826289828009491476476440647, 16.624610400393656045596332757168, 17.4061081001285612944787984350, 17.613879034599868318040112578559, 18.61829794211516167116316795879

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