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Properties

Label 1-3744-3744.421-r1-0-0
Degree $1$
Conductor $3744$
Sign $0.744 - 0.667i$
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 + (0.5 + 0.866i)7-s + (−0.965 − 0.258i)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 + (0.866 + 0.5i)31-s + (−0.707 + 0.707i)35-s + (−0.707 + 0.707i)37-s + (0.5 − 0.866i)41-s + (0.258 − 0.965i)43-s + (−0.866 + 0.5i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)5-s + (0.5 + 0.866i)7-s + (−0.965 − 0.258i)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 + (0.866 + 0.5i)31-s + (−0.707 + 0.707i)35-s + (−0.707 + 0.707i)37-s + (0.5 − 0.866i)41-s + (0.258 − 0.965i)43-s + (−0.866 + 0.5i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.744 - 0.667i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ 0.744 - 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.320830639 - 0.5057629704i\)
\(L(\frac12)\) \(\approx\) \(1.320830639 - 0.5057629704i\)
\(L(1)\) \(\approx\) \(1.015253756 + 0.2014794286i\)
\(L(1)\) \(\approx\) \(1.015253756 + 0.2014794286i\)

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

−18.36269329750275768143733627186, −17.74604119369628899226130470681, −17.16822878400151126201280567659, −16.47385521337233477540480125003, −15.98322975757860366707548524506, −15.11981815025728588756781988762, −14.23769424665346587243620901348, −13.74759741813633634552682771732, −12.99571574839335170101915936202, −12.39531239010239389754648018744, −11.715912733736703180757357771706, −10.77730262127627415959968714389, −10.04241340474508103988690918247, −9.70416078817073095438242353277, −8.49361328270295021454254887285, −7.97532004522584619577411328905, −7.53760992418371051679276656903, −6.32619470299834823595085321435, −5.66918653341317908651987535932, −4.81418216230616046572630843556, −4.338107314951412515695813295779, −3.4337583158936340280330370996, −2.31453212540112977298647282232, −1.494293346728720540435549785972, −0.75652710387078160891872684424, 0.24849408769928686075283088422, 1.55479221191779839600618307868, 2.49346976522776387301743065583, 2.846237770589457843017599004589, 3.85326661524406482299073864076, 4.97834058623513243330697684837, 5.47296932014598354584452845190, 6.360131685481550001538637400187, 6.9097512266848667123569242455, 8.11793458712134223855523422292, 8.2183891221715767443652043671, 9.35245065384585820623958874140, 10.167753385508789949826405034, 10.66068312067796545530780137712, 11.371679036367927962298038572459, 12.17815585101429795538723073705, 12.72285938271332815693342042108, 13.851546087627568286315537903937, 14.14422727235126057411415813252, 14.99039468223338542754737265638, 15.57478862513389143769588790556, 16.10234886415898771141226059441, 17.22164648351724452261864968590, 17.81410061910732252154013976663, 18.32075621626274127825790448614

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