Properties

Label 1-4235-4235.628-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.476 + 0.879i$
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  + (0.945 + 0.327i)2-s + (0.866 + 0.5i)3-s + (0.786 + 0.618i)4-s + (0.654 + 0.755i)6-s + (0.540 + 0.841i)8-s + (0.5 + 0.866i)9-s + (0.371 + 0.928i)12-s + (0.989 + 0.142i)13-s + (0.235 + 0.971i)16-s + (0.0950 − 0.995i)17-s + (0.189 + 0.981i)18-s + (−0.995 + 0.0950i)19-s + (−0.971 + 0.235i)23-s + (0.0475 + 0.998i)24-s + (0.888 + 0.458i)26-s + i·27-s + ⋯
L(s)  = 1  + (0.945 + 0.327i)2-s + (0.866 + 0.5i)3-s + (0.786 + 0.618i)4-s + (0.654 + 0.755i)6-s + (0.540 + 0.841i)8-s + (0.5 + 0.866i)9-s + (0.371 + 0.928i)12-s + (0.989 + 0.142i)13-s + (0.235 + 0.971i)16-s + (0.0950 − 0.995i)17-s + (0.189 + 0.981i)18-s + (−0.995 + 0.0950i)19-s + (−0.971 + 0.235i)23-s + (0.0475 + 0.998i)24-s + (0.888 + 0.458i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.476 + 0.879i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (628, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.476 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.400230680 + 4.031084261i\)
\(L(\frac12)\) \(\approx\) \(2.400230680 + 4.031084261i\)
\(L(1)\) \(\approx\) \(2.147994365 + 1.305044728i\)
\(L(1)\) \(\approx\) \(2.147994365 + 1.305044728i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.945 - 0.327i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-0.989 - 0.142i)T \)
17 \( 1 + (-0.0950 + 0.995i)T \)
19 \( 1 + (0.995 - 0.0950i)T \)
23 \( 1 + (0.971 - 0.235i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (0.928 + 0.371i)T \)
37 \( 1 + (-0.618 - 0.786i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (-0.540 - 0.841i)T \)
47 \( 1 + (0.189 - 0.981i)T \)
53 \( 1 + (-0.971 - 0.235i)T \)
59 \( 1 + (0.327 + 0.945i)T \)
61 \( 1 + (0.981 + 0.189i)T \)
67 \( 1 + (0.189 + 0.981i)T \)
71 \( 1 + (-0.415 + 0.909i)T \)
73 \( 1 + (0.690 + 0.723i)T \)
79 \( 1 + (0.0475 - 0.998i)T \)
83 \( 1 + (-0.281 - 0.959i)T \)
89 \( 1 + (-0.580 + 0.814i)T \)
97 \( 1 + (-0.540 - 0.841i)T \)
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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

−18.39305316784704721002805407290, −17.612305946077202748643244163258, −16.606319583701531867963324464865, −15.8578363300870856323657617085, −15.1706395773666872783141808533, −14.6873421920254897219292783792, −14.00372871098472788562241335714, −13.3479037275598709570964295199, −12.845183556664822665663920739057, −12.27599232685438603814090789974, −11.465769881117115871225282828311, −10.60953823527825373479552985268, −10.12838763775265659674583359334, −9.04293810967186483487608926332, −8.47212819969563657118845562141, −7.60031452083407028397157170762, −6.94549193125621136387513744025, −5.96625860566312319226429759672, −5.787742548428740581166545102599, −4.18475572946433804207374128305, −4.032417807500075615747768481702, −3.19437811494495372011902636663, −2.168570270895623182731431252089, −1.84714739543043221941853961286, −0.731190866660451641817131668, 1.45574508687582721373281895979, 2.2482693664188827651825903996, 3.05614611499888376805111416851, 3.70107917072414093174668245166, 4.38033884406502420086480782833, 5.0110425205483857916318400749, 5.97586794113621156982119962256, 6.570224271332721567587666803252, 7.66768702473326498442618003996, 7.92912909195478243288105226747, 8.92780296753409397751617185225, 9.46816264896546632334182624452, 10.571201478725804939601892992956, 11.047818368963929809592915024998, 11.83194674273406046830500280893, 12.837764945608002940245684813750, 13.20516867194826147485084056719, 14.05694220603210467757636209294, 14.39581693755921065515818865018, 15.17458225125253549000551844693, 15.74736069094585029151672821375, 16.39737290989583002254284190040, 16.78154110125144729375267414361, 18.01955212205291110851096675962, 18.557366410519365680190594744645

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