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Properties

Label 1-3744-3744.1883-r1-0-0
Degree $1$
Conductor $3744$
Sign $0.446 + 0.894i$
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 + (−0.707 + 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 + (0.866 + 0.5i)31-s + (0.965 + 0.258i)35-s + (0.258 + 0.965i)37-s + (0.5 + 0.866i)41-s + (0.965 − 0.258i)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 + (−0.707 + 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 + (0.866 + 0.5i)31-s + (0.965 + 0.258i)35-s + (0.258 + 0.965i)37-s + (0.5 + 0.866i)41-s + (0.965 − 0.258i)43-s + (0.866 − 0.5i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.446 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.446 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.446 + 0.894i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ 0.446 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.462551717 + 1.522756180i\)
\(L(\frac12)\) \(\approx\) \(2.462551717 + 1.522756180i\)
\(L(1)\) \(\approx\) \(1.237063078 + 0.2641954027i\)
\(L(1)\) \(\approx\) \(1.237063078 + 0.2641954027i\)

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 \( 1 + (-0.707 + 0.707i)T \)
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 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (0.258 + 0.965i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.965 - 0.258i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (-0.258 - 0.965i)T \)
67 \( 1 + (0.258 - 0.965i)T \)
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 \( 1 + (-0.866 - 0.5i)T \)
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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

−18.12164602052446340637817916636, −17.76084584665669403544592211539, −16.97094130555221239463165321893, −16.12398448433671890458046216638, −15.77073752919101501886113910244, −14.99083742215728985804334863432, −14.077340259674615027036286236716, −13.50467122745055954333228455859, −12.78445617562319269893566572133, −12.16482723148852663476196695967, −11.43310266406923112300150527422, −10.808469078794022352431368572000, −9.82555706353181476687267241782, −8.98337560493644877842889572407, −8.74019328858515630376042877365, −7.85764215853712822848579438492, −7.13423479545498723892607972204, −5.95714087609060514949796682859, −5.419363300213675161240013556961, −4.914792470108605337853220699798, −4.05774160675701340923180731067, −2.75938104351014426897376821318, −2.427450493157710649047745706598, −1.12867249294411100872830738020, −0.58691597786786528131383459837, 0.77261058817750955228889097794, 1.70513552689081000085315661906, 2.52588699228254518285494550402, 3.3366822487074878280007683342, 4.213129861048598684080953432488, 4.86851023376254690919869061424, 5.909301045793843157675835722457, 6.52754148673349822213430515664, 7.36572212074103746943695722968, 7.86168447902065374508445164083, 8.60189533986902785571880948890, 9.90168791692438863654462547947, 10.226009420356219601227369565933, 10.7422150745975927844447095173, 11.5418582045653998364298520478, 12.4043206762365008462702942561, 13.109392589430050918791236349343, 13.89938851677704132739362733860, 14.445054170616412687136651178999, 15.047505490435213834755945406735, 15.66998080688046308998023033969, 16.71274954429309462592373579332, 17.302489242681515372257957734946, 17.770667250522235960570817325341, 18.67544686949537000100991669590

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