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Properties

Label 1-3744-3744.29-r1-0-0
Degree $1$
Conductor $3744$
Sign $0.408 - 0.912i$
Analytic cond. $402.348$
Root an. cond. $402.348$
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  + (0.258 + 0.965i)5-s + (0.866 + 0.5i)7-s + (−0.707 + 0.707i)11-s + (−0.5 − 0.866i)17-s + (−0.258 + 0.965i)19-s + (−0.866 + 0.5i)23-s + (−0.866 + 0.5i)25-s + (−0.707 − 0.707i)29-s + (−0.5 + 0.866i)31-s + (−0.258 + 0.965i)35-s + (−0.258 − 0.965i)37-s + (−0.866 + 0.5i)41-s + (0.258 + 0.965i)43-s + (−0.5 − 0.866i)47-s + (0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)5-s + (0.866 + 0.5i)7-s + (−0.707 + 0.707i)11-s + (−0.5 − 0.866i)17-s + (−0.258 + 0.965i)19-s + (−0.866 + 0.5i)23-s + (−0.866 + 0.5i)25-s + (−0.707 − 0.707i)29-s + (−0.5 + 0.866i)31-s + (−0.258 + 0.965i)35-s + (−0.258 − 0.965i)37-s + (−0.866 + 0.5i)41-s + (0.258 + 0.965i)43-s + (−0.5 − 0.866i)47-s + (0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.408 - 0.912i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ 0.408 - 0.912i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3380554798 - 0.2190438402i\)
\(L(\frac12)\) \(\approx\) \(0.3380554798 - 0.2190438402i\)
\(L(1)\) \(\approx\) \(0.9030239501 + 0.3023166398i\)
\(L(1)\) \(\approx\) \(0.9030239501 + 0.3023166398i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.258 + 0.965i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.258 + 0.965i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (-0.965 + 0.258i)T \)
67 \( 1 + (0.258 - 0.965i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.258 - 0.965i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−18.44762149360553704939828539886, −17.81226097218077306820550525966, −17.11540973945762522102268353141, −16.69167245373976740278500296576, −15.86178864438712286020978004952, −15.21554212862985875673045016060, −14.39421048356716827115714782454, −13.58206671695131010505292372130, −13.20249133879933319488527769607, −12.461137378396167748331885653251, −11.59001952841334586902273290018, −10.90807866526485752812551022379, −10.35268435966555564934032050500, −9.41283165418719760962106708259, −8.547924477958509186053851121798, −8.26097280134862911531313521074, −7.42121018240428577377776770260, −6.45100472128898548643078465316, −5.61831106883222859552487996801, −4.97988723183918297618852011830, −4.31366436877348904789530453881, −3.536352551542136736378370269, −2.28552812955436235919446814661, −1.68214354327710582797748147617, −0.69490144131253985147197216043, 0.07063792597445333542649109675, 1.72458710437433714910668442414, 2.07409006949122950787002052728, 2.964859160183942100100070475785, 3.86233376807751011444363738084, 4.79968734220765136779148465570, 5.49310201505217477404656560469, 6.19900697517454773021151260645, 7.119899920552935853908717657739, 7.71168153591662020162478735289, 8.33894328140800191267562633097, 9.42044170497570985678763517639, 9.929198746918965792778085960374, 10.773891115333621593086783051696, 11.29657657356641309025898702586, 12.03811352850811369637072499422, 12.75372310952913313249755109957, 13.71240600969508047195553985464, 14.22416972192166215825658542081, 14.91697525816080564934664342551, 15.442918427701819048096768776172, 16.13072301174447875978152357084, 17.143935237323825561299645524449, 17.87805979574386330051650544903, 18.2528845120130659610746899939

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