Properties

Degree $2$
Conductor $5376$
Sign $-0.707 + 0.707i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  i·3-s − 7-s − 9-s − 6i·11-s − 2i·13-s + 4i·19-s + i·21-s + 6·23-s + 5·25-s + i·27-s − 6i·29-s + 8·31-s − 6·33-s + 2i·37-s − 2·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.377·7-s − 0.333·9-s − 1.80i·11-s − 0.554i·13-s + 0.917i·19-s + 0.218i·21-s + 1.25·23-s + 25-s + 0.192i·27-s − 1.11i·29-s + 1.43·31-s − 1.04·33-s + 0.328i·37-s − 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5376\)    =    \(2^{8} \cdot 3 \cdot 7\)
Sign: $-0.707 + 0.707i$
Motivic weight: \(1\)
Character: $\chi_{5376} (2689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5376,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.548788080\)
\(L(\frac12)\) \(\approx\) \(1.548788080\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
7 \( 1 + T \)
good5 \( 1 - 5T^{2} \)
11 \( 1 + 6iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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

−8.197451565090073380256223845883, −7.07160129917770707532353241715, −6.52019958098699397443596794088, −5.76968604103250793534544153391, −5.27091455477345115090835253257, −4.07276950725577530288946453273, −3.15676367888424551467572689747, −2.71297757063755141625942169239, −1.26984996622580969784215597678, −0.46189172833971569665476868801, 1.20854458323223828249294801970, 2.41287830004445153178061741925, 3.10707592291113895267410181114, 4.18499098222412554668901955667, 4.79288320765838573004329877568, 5.25626809348525549333895544894, 6.59254268668578288248080930839, 6.87401811512946843816273819919, 7.59675164895588334070348386317, 8.741376925226365980698439582215

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