↑

Properties

Label 2-1224-1224.515-c0-0-0
Degree $2$
Conductor $1224$
Sign $0.206 - 0.978i$
Analytic cond. $0.610855$
Root an. cond. $0.781572$
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 + (0.991 + 0.130i)3-s + (−0.965 − 0.258i)4-s + (−0.258 + 0.965i)6-s + (0.382 − 0.923i)8-s + (0.965 + 0.258i)9-s + (−0.0578 − 0.117i)11-s + (−0.923 − 0.382i)12-s + (0.866 + 0.5i)16-s + (0.608 − 0.793i)17-s + (−0.382 + 0.923i)18-s + (0.739 + 1.78i)19-s + (0.123 − 0.0420i)22-s + (0.5 − 0.866i)24-s + (−0.793 + 0.608i)25-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (0.991 + 0.130i)3-s + (−0.965 − 0.258i)4-s + (−0.258 + 0.965i)6-s + (0.382 − 0.923i)8-s + (0.965 + 0.258i)9-s + (−0.0578 − 0.117i)11-s + (−0.923 − 0.382i)12-s + (0.866 + 0.5i)16-s + (0.608 − 0.793i)17-s + (−0.382 + 0.923i)18-s + (0.739 + 1.78i)19-s + (0.123 − 0.0420i)22-s + (0.5 − 0.866i)24-s + (−0.793 + 0.608i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.206 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.295512832\)
\(L(\frac12)\) \(\approx\) \(1.295512832\)
\(L(1)\) 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 \( 1 + (-0.991 - 0.130i)T \)
17 \( 1 + (-0.608 + 0.793i)T \)
good5 \( 1 + (0.793 - 0.608i)T^{2} \)
7 \( 1 + (-0.793 - 0.608i)T^{2} \)
11 \( 1 + (0.0578 + 0.117i)T + (-0.608 + 0.793i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.739 - 1.78i)T + (-0.707 + 0.707i)T^{2} \)
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 \( 1 + (1.31 + 1.50i)T + (-0.130 + 0.991i)T^{2} \)
43 \( 1 + (0.741 + 0.965i)T + (-0.258 + 0.965i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.258 + 0.0340i)T + (0.965 - 0.258i)T^{2} \)
61 \( 1 + (0.793 + 0.608i)T^{2} \)
67 \( 1 + (1.37 - 0.793i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.382 + 0.923i)T^{2} \)
73 \( 1 + (-0.172 + 0.867i)T + (-0.923 - 0.382i)T^{2} \)
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 \( 1 + (-0.991 + 1.13i)T + (-0.130 - 0.991i)T^{2} \)
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

−9.968628203046819118382864804446, −9.080998506726375958431379823565, −8.421568370156410314557594419969, −7.57336200572014890763822780591, −7.18173994267915470688768803744, −5.88904535071486304451085906428, −5.16719320117643873570215025064, −3.98193916461049155017443002720, −3.27731426600377709206052513393, −1.60964664061964359731485009065, 1.36133624318732821079596927104, 2.53200576275795315873008682711, 3.31858626429759564719699486054, 4.28070034350693345658547639712, 5.17465671014218782949909761886, 6.58202237038833111172480724610, 7.64327991789084615392508435128, 8.286659135915719940776013280192, 9.046542854076860664164623611213, 9.785008326618562371883244296943

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