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Properties

Label 1-3744-3744.77-r1-0-0
Degree $1$
Conductor $3744$
Sign $-0.854 + 0.518i$
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  + (−0.965 − 0.258i)5-s + (−0.866 − 0.5i)7-s + (−0.258 − 0.965i)11-s + 17-s + (−0.707 − 0.707i)19-s + (−0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s + (−0.965 + 0.258i)29-s + (0.5 + 0.866i)31-s + (0.707 + 0.707i)35-s + (−0.707 + 0.707i)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  + (−0.965 − 0.258i)5-s + (−0.866 − 0.5i)7-s + (−0.258 − 0.965i)11-s + 17-s + (−0.707 − 0.707i)19-s + (−0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s + (−0.965 + 0.258i)29-s + (0.5 + 0.866i)31-s + (0.707 + 0.707i)35-s + (−0.707 + 0.707i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.854 + 0.518i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ -0.854 + 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1832016725 - 0.6550510300i\)
\(L(\frac12)\) \(\approx\) \(-0.1832016725 - 0.6550510300i\)
\(L(1)\) \(\approx\) \(0.6830653448 - 0.2338104325i\)
\(L(1)\) \(\approx\) \(0.6830653448 - 0.2338104325i\)

Euler product

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

−18.76525275503815565213570703016, −18.47952472873096753276233691620, −17.46093766389768401185472567740, −16.637629762604872815788045042854, −16.06176021996240396549310095092, −15.45914038046619299269495841931, −14.78300220941484971858098684135, −14.297205229940072227060845440348, −13.092270151956923884103052932846, −12.5264098177116072047233863602, −12.10493573895148477639453526940, −11.322956214793029495583562131055, −10.38929508193674106923461485382, −9.87833399940484685884246834374, −9.120575228864040665243599187665, −8.12856306950135395785328870674, −7.68447658980316655062363628993, −6.87698260385272291691670328191, −6.09128941066715684264030241780, −5.40689702221845903628354879560, −4.18959784957582414137397270030, −3.91005535883115401782366901854, −2.81715925658372977199589833662, −2.23830659414718726524011146169, −0.965003148363379242089562436454, 0.190414015743184654753167440499, 0.57844730796926896239066641718, 1.78216594863120275397496503965, 3.10098276225711241551977236149, 3.47453685617576457899035008436, 4.215086628699409463522818170377, 5.1869147394140756176621169194, 5.921842264818927921498519557845, 6.838529911622135151762841685481, 7.42614365142915348502116803094, 8.2367082068523602792671446501, 8.81175059921313126153504459142, 9.67433198653715867614421029087, 10.520852395053184536106661748524, 11.00788419816060443749075746199, 11.97329954873489704942839212893, 12.39148053520595883919521695469, 13.30029657524174488744533048586, 13.77093719661481392863378896023, 14.657639355682387833012548348281, 15.55595454063204705043620952448, 15.92251306524554529082905066965, 16.69929140438146309941523177681, 17.05688593925556091591508308845, 18.1859148548388018696528193848

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