↑

Properties

Label 2-3672-1224.707-c0-0-1
Degree $2$
Conductor $3672$
Sign $0.442 - 0.896i$
Analytic cond. $1.83256$
Root an. cond. $1.35372$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.965 + 0.258i)4-s + (−0.382 − 0.923i)8-s + (−0.793 − 0.391i)11-s + (0.866 − 0.5i)16-s + (0.991 − 0.130i)17-s + (0.198 − 0.478i)19-s + (0.284 − 0.837i)22-s + (0.793 + 0.608i)25-s + (0.608 + 0.793i)32-s + (0.258 + 0.965i)34-s + (0.5 + 0.133i)38-s + (1.34 + 1.18i)41-s + (1.20 − 1.57i)43-s + (0.867 + 0.172i)44-s + ⋯
L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.965 + 0.258i)4-s + (−0.382 − 0.923i)8-s + (−0.793 − 0.391i)11-s + (0.866 − 0.5i)16-s + (0.991 − 0.130i)17-s + (0.198 − 0.478i)19-s + (0.284 − 0.837i)22-s + (0.793 + 0.608i)25-s + (0.608 + 0.793i)32-s + (0.258 + 0.965i)34-s + (0.5 + 0.133i)38-s + (1.34 + 1.18i)41-s + (1.20 − 1.57i)43-s + (0.867 + 0.172i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3672\)    =    \(2^{3} \cdot 3^{3} \cdot 17\)
Sign: $0.442 - 0.896i$
Analytic conductor: \(1.83256\)
Root analytic conductor: \(1.35372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3672} (2339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3672,\ (\ :0),\ 0.442 - 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.189337612\)
\(L(\frac12)\) \(\approx\) \(1.189337612\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.130 - 0.991i)T \)
3 \( 1 \)
17 \( 1 + (-0.991 + 0.130i)T \)
good5 \( 1 + (-0.793 - 0.608i)T^{2} \)
7 \( 1 + (0.793 - 0.608i)T^{2} \)
11 \( 1 + (0.793 + 0.391i)T + (0.608 + 0.793i)T^{2} \)
13 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.198 + 0.478i)T + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.991 + 0.130i)T^{2} \)
29 \( 1 + (0.130 - 0.991i)T^{2} \)
31 \( 1 + (0.608 - 0.793i)T^{2} \)
37 \( 1 + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (-1.34 - 1.18i)T + (0.130 + 0.991i)T^{2} \)
43 \( 1 + (-1.20 + 1.57i)T + (-0.258 - 0.965i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-1.57 - 0.207i)T + (0.965 + 0.258i)T^{2} \)
61 \( 1 + (-0.793 + 0.608i)T^{2} \)
67 \( 1 + (-0.226 - 0.130i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.382 + 0.923i)T^{2} \)
73 \( 1 + (0.128 - 0.0255i)T + (0.923 - 0.382i)T^{2} \)
79 \( 1 + (0.608 + 0.793i)T^{2} \)
83 \( 1 + (0.758 - 0.0999i)T + (0.965 - 0.258i)T^{2} \)
89 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
97 \( 1 + (0.483 - 0.423i)T + (0.130 - 0.991i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.735602376497353752432047634003, −7.902996425253391132736978343014, −7.44833607268876568747106099555, −6.66863273195653770636487491703, −5.75698097466258495772563117917, −5.29312039470910839178192007073, −4.46220922282686568081872458721, −3.48671708079967047303230375116, −2.67859600297468537768071128948, −0.924926275774336726131122430564, 0.972495576702502472421378714593, 2.17718508539955666237986407245, 2.95175205262374725163672918075, 3.83893456196690157343878800395, 4.67453697321803867297753387816, 5.41883335062205380584295862313, 6.09735769949246585387314703912, 7.28824427361583428148603535237, 8.002151873319001580139563644038, 8.632135466966061222975246797321

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