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Properties

Label 2-1224-1224.331-c0-0-1
Degree $2$
Conductor $1224$
Sign $-0.582 + 0.812i$
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  + (0.258 + 0.965i)2-s + (−0.866 − 0.5i)3-s + (−0.866 + 0.499i)4-s + (0.258 − 0.965i)6-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (−1.96 − 0.258i)11-s + 12-s + (0.500 − 0.866i)16-s + (−0.258 − 0.965i)17-s + (−0.707 + 0.707i)18-s + (−1.22 + 1.22i)19-s + (−0.258 − 1.96i)22-s + (0.258 + 0.965i)24-s + (−0.965 + 0.258i)25-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 − 0.5i)3-s + (−0.866 + 0.499i)4-s + (0.258 − 0.965i)6-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (−1.96 − 0.258i)11-s + 12-s + (0.500 − 0.866i)16-s + (−0.258 − 0.965i)17-s + (−0.707 + 0.707i)18-s + (−1.22 + 1.22i)19-s + (−0.258 − 1.96i)22-s + (0.258 + 0.965i)24-s + (−0.965 + 0.258i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.582 + 0.812i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ -0.582 + 0.812i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01067729722\)
\(L(\frac12)\) \(\approx\) \(0.01067729722\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (0.258 + 0.965i)T \)
good5 \( 1 + (0.965 - 0.258i)T^{2} \)
7 \( 1 + (0.965 + 0.258i)T^{2} \)
11 \( 1 + (1.96 + 0.258i)T + (0.965 + 0.258i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
23 \( 1 + (0.258 + 0.965i)T^{2} \)
29 \( 1 + (-0.258 + 0.965i)T^{2} \)
31 \( 1 + (-0.965 + 0.258i)T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (1.57 + 1.20i)T + (0.258 + 0.965i)T^{2} \)
43 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.965 + 0.258i)T^{2} \)
67 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (-1.12 - 0.465i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.965 - 0.258i)T^{2} \)
83 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + (0.207 - 0.158i)T + (0.258 - 0.965i)T^{2} \)
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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

−9.716696540408937355131299657036, −8.275454501830412813846413552979, −7.996354978610156993505522666266, −7.05533947851699776261451697206, −6.30730306869633454340799234976, −5.36050059636391010543680762478, −5.01919991266248048839503339961, −3.72241139277576827031375825266, −2.25789506771108641155942810402, −0.008838924420423837138712630765, 1.98450103875216767823258243688, 3.11580570595110078650162082915, 4.32685756189131148015430243577, 4.91048550078637219776184184332, 5.73909021635789191189936512799, 6.61940439889836521560128636530, 7.970260519728575942506718479460, 8.767070054980905889949715876672, 9.837586463312813286848508434243, 10.37320347058373342119005918658

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