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Properties

Label 1-3744-3744.3035-r1-0-0
Degree $1$
Conductor $3744$
Sign $-0.999 + 0.0167i$
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 + 7-s + (0.258 + 0.965i)11-s + (0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s + i·23-s + (−0.866 + 0.5i)25-s + (0.258 + 0.965i)29-s + (−0.866 − 0.5i)31-s + (−0.258 − 0.965i)35-s + (0.965 + 0.258i)37-s + 41-s + (−0.707 − 0.707i)43-s + (−0.866 + 0.5i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)5-s + 7-s + (0.258 + 0.965i)11-s + (0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s + i·23-s + (−0.866 + 0.5i)25-s + (0.258 + 0.965i)29-s + (−0.866 − 0.5i)31-s + (−0.258 − 0.965i)35-s + (0.965 + 0.258i)37-s + 41-s + (−0.707 − 0.707i)43-s + (−0.866 + 0.5i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.999 + 0.0167i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3035, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ -0.999 + 0.0167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001817725844 - 0.2164277388i\)
\(L(\frac12)\) \(\approx\) \(0.001817725844 - 0.2164277388i\)
\(L(1)\) \(\approx\) \(1.007628878 - 0.09698561171i\)
\(L(1)\) \(\approx\) \(1.007628878 - 0.09698561171i\)

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 \( 1 + T \)
11 \( 1 + (0.258 + 0.965i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + iT \)
29 \( 1 + (0.258 + 0.965i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.965 + 0.258i)T \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.258 + 0.965i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - iT \)
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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

−18.78991603361432516631076645359, −18.12287388119868021628638447390, −17.54166511235744585100128892818, −16.71253796223495591588867383287, −16.150712291135836348177545303799, −14.97579525484870013417267908681, −14.813651965537838963578991295096, −14.184965887686706434302944208, −13.37878567609154762991159992328, −12.547243599659211184875863600920, −11.67493578189922117776912797264, −11.100852297154754536715947485538, −10.66164620174403731979158345138, −9.90088830247416759643592913060, −8.79120781674388901305873278322, −8.19705395485484342711993693556, −7.68292413664171868849545789953, −6.62345581533328560295185483619, −6.17166527849970694867761026612, −5.2886295458247032692517319752, −4.26360277854351368151050846261, −3.741704488805076318718599114626, −2.74432689220451320327104405010, −2.05222921837066120810275548837, −1.01859964119508633026705165791, 0.03310675712484179367547528410, 1.22173573588484344854155409077, 1.68301991660807784621678412578, 2.69892024214372542608105932, 3.94518469556838064969230111937, 4.42989772460165249189208317481, 5.18587712186410819877114838484, 5.744894421783163786491012826953, 6.98099200416137912510300152724, 7.601295923076295448855095016505, 8.20938486580200838334619072858, 9.043079758279959708308741740816, 9.55219450830552215798152791842, 10.46312960580435434983577477724, 11.346459015255814245551121312973, 11.87944670284687928332741980323, 12.53217480504087782218672126864, 13.18762073873570525829732593499, 14.02070331987967347770835493918, 14.80577490400523297025734112497, 15.21543982691219913214177709026, 16.19326786699783141781926195277, 16.73011468245070822427102110131, 17.415245734780761778301304042872, 18.00635403631747926659047936342

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