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Properties

Label 2-1224-1224.227-c0-0-0
Degree $2$
Conductor $1224$
Sign $-0.986 + 0.162i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.608 + 0.793i)2-s + (−0.793 + 0.608i)3-s + (−0.258 + 0.965i)4-s + (−0.965 − 0.258i)6-s + (−0.923 + 0.382i)8-s + (0.258 − 0.965i)9-s + (−1.24 + 1.42i)11-s + (−0.382 − 0.923i)12-s + (−0.866 − 0.499i)16-s + (0.130 + 0.991i)17-s + (0.923 − 0.382i)18-s + (−0.478 − 0.198i)19-s + (−1.88 − 0.123i)22-s + (0.499 − 0.866i)24-s + (−0.991 − 0.130i)25-s + ⋯
L(s)  = 1  + (0.608 + 0.793i)2-s + (−0.793 + 0.608i)3-s + (−0.258 + 0.965i)4-s + (−0.965 − 0.258i)6-s + (−0.923 + 0.382i)8-s + (0.258 − 0.965i)9-s + (−1.24 + 1.42i)11-s + (−0.382 − 0.923i)12-s + (−0.866 − 0.499i)16-s + (0.130 + 0.991i)17-s + (0.923 − 0.382i)18-s + (−0.478 − 0.198i)19-s + (−1.88 − 0.123i)22-s + (0.499 − 0.866i)24-s + (−0.991 − 0.130i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.986 + 0.162i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ -0.986 + 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7685454623\)
\(L(\frac12)\) \(\approx\) \(0.7685454623\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.608 - 0.793i)T \)
3 \( 1 + (0.793 - 0.608i)T \)
17 \( 1 + (-0.130 - 0.991i)T \)
good5 \( 1 + (0.991 + 0.130i)T^{2} \)
7 \( 1 + (-0.991 + 0.130i)T^{2} \)
11 \( 1 + (1.24 - 1.42i)T + (-0.130 - 0.991i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.478 + 0.198i)T + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.793 + 0.608i)T^{2} \)
29 \( 1 + (0.608 - 0.793i)T^{2} \)
31 \( 1 + (-0.130 + 0.991i)T^{2} \)
37 \( 1 + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (-0.576 - 0.284i)T + (0.608 + 0.793i)T^{2} \)
43 \( 1 + (0.0340 - 0.258i)T + (-0.965 - 0.258i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.965 - 0.741i)T + (0.258 + 0.965i)T^{2} \)
61 \( 1 + (0.991 - 0.130i)T^{2} \)
67 \( 1 + (-1.71 + 0.991i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.732 - 1.09i)T + (-0.382 - 0.923i)T^{2} \)
79 \( 1 + (-0.130 - 0.991i)T^{2} \)
83 \( 1 + (1.46 - 1.12i)T + (0.258 - 0.965i)T^{2} \)
89 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
97 \( 1 + (0.793 - 0.391i)T + (0.608 - 0.793i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.25760336691234202613568163338, −9.669035779793884589093513854348, −8.576717598127787545893601863733, −7.69757669651280188893892483488, −6.93366553314498659716109934524, −6.01742281788096845020296834330, −5.31192954301789386183499902905, −4.51065635407557995124409933771, −3.81218481551222060776734760279, −2.37413522639109099777356112612, 0.58101781224752806854027485419, 2.13512939875867708074888504415, 3.10086550017742352685685532978, 4.34774460475752411281116657525, 5.49316900955862074686576143275, 5.70240370513536608856422742692, 6.80682501464358362435074960487, 7.81822040521701375582238982108, 8.695613559172773103169718125157, 9.860018102756811129542056318003

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