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Properties

Label 1-4235-4235.788-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.997 - 0.0638i$
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  + (0.703 − 0.710i)2-s + (−0.994 − 0.104i)3-s + (−0.00951 − 0.999i)4-s + (−0.774 + 0.633i)6-s + (−0.717 − 0.696i)8-s + (0.978 + 0.207i)9-s + (−0.0950 + 0.995i)12-s + (0.676 + 0.736i)13-s + (−0.999 + 0.0190i)16-s + (0.475 + 0.879i)17-s + (0.836 − 0.548i)18-s + (−0.991 + 0.132i)19-s + (0.945 − 0.327i)23-s + (0.640 + 0.768i)24-s + (0.999 + 0.0380i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (0.703 − 0.710i)2-s + (−0.994 − 0.104i)3-s + (−0.00951 − 0.999i)4-s + (−0.774 + 0.633i)6-s + (−0.717 − 0.696i)8-s + (0.978 + 0.207i)9-s + (−0.0950 + 0.995i)12-s + (0.676 + 0.736i)13-s + (−0.999 + 0.0190i)16-s + (0.475 + 0.879i)17-s + (0.836 − 0.548i)18-s + (−0.991 + 0.132i)19-s + (0.945 − 0.327i)23-s + (0.640 + 0.768i)24-s + (0.999 + 0.0380i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.997 - 0.0638i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (788, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.997 - 0.0638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.483914456 - 0.04744606989i\)
\(L(\frac12)\) \(\approx\) \(1.483914456 - 0.04744606989i\)
\(L(1)\) \(\approx\) \(1.027490123 - 0.3900086342i\)
\(L(1)\) \(\approx\) \(1.027490123 - 0.3900086342i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.703 - 0.710i)T \)
3 \( 1 + (-0.994 - 0.104i)T \)
13 \( 1 + (0.676 + 0.736i)T \)
17 \( 1 + (0.475 + 0.879i)T \)
19 \( 1 + (-0.991 + 0.132i)T \)
23 \( 1 + (0.945 - 0.327i)T \)
29 \( 1 + (0.941 - 0.336i)T \)
31 \( 1 + (-0.398 - 0.917i)T \)
37 \( 1 + (-0.318 + 0.948i)T \)
41 \( 1 + (0.362 + 0.931i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 + (-0.836 - 0.548i)T \)
53 \( 1 + (0.0190 - 0.999i)T \)
59 \( 1 + (-0.988 + 0.151i)T \)
61 \( 1 + (-0.964 - 0.263i)T \)
67 \( 1 + (-0.998 - 0.0475i)T \)
71 \( 1 + (-0.564 + 0.825i)T \)
73 \( 1 + (0.424 + 0.905i)T \)
79 \( 1 + (-0.0665 + 0.997i)T \)
83 \( 1 + (0.389 + 0.921i)T \)
89 \( 1 + (-0.235 + 0.971i)T \)
97 \( 1 + (0.170 + 0.985i)T \)
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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

−17.92662921173339539324255974290, −17.721303858065482608869403512279, −16.87564116426058402438244820057, −16.30347829363645434286223538873, −15.6860811709898708985573479551, −15.22686050642192280141723664294, −14.30242607181494566883769713668, −13.660168037674655885147360874008, −12.771934860645513655595518296764, −12.47625387366485182507031966238, −11.69248289160421302457146795307, −10.84848898095877175406153515368, −10.48404094678745920982000973966, −9.20006230610088475025571508694, −8.71607419782646487431190281841, −7.494307680842470369348221357142, −7.26109157789108365214967277774, −6.16560011446235097986887394817, −5.89901187783896278352962292176, −4.92990498489137316964315351877, −4.573992888985110260499828394700, −3.518719641165648575770771717993, −2.904970871959358847439455402680, −1.60189581818757111398245008328, −0.43361015214249071410170621759, 0.95061819568929237463413403806, 1.58882702151623382278212496005, 2.470609890474328043496772604849, 3.557117179503504201242915554048, 4.27181756253594207053925082280, 4.80466163330320649094253968308, 5.70998946695550611406011274314, 6.360585718731979070779143311961, 6.72196236485136879484361148137, 7.92401954702401454694058269806, 8.84289806219773838979258179629, 9.7326678828547755522390696570, 10.381974482862943744395876638630, 11.013730628911619844024616465200, 11.48884647508743175399434359284, 12.249416745015749319350915273117, 12.84536997614369130201654866637, 13.34321521848958378530376981377, 14.19096312267661734395041793304, 14.9929526709031415126294781407, 15.50877417991573986571602538338, 16.47002657083118817558977359553, 16.90219196794653497499011526577, 17.800573531817566133354583685138, 18.51837338201551309984986975081

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