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Properties

Label 2-1224-1224.347-c0-0-0
Degree $2$
Conductor $1224$
Sign $-0.452 - 0.891i$
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  + (0.130 + 0.991i)2-s + (0.965 + 0.258i)3-s + (−0.965 + 0.258i)4-s + (−0.130 + 0.991i)6-s + (−0.382 − 0.923i)8-s + (0.866 + 0.499i)9-s + (−0.793 + 1.60i)11-s − 12-s + (0.866 − 0.5i)16-s + (0.991 − 0.130i)17-s + (−0.382 + 0.923i)18-s + (−0.198 + 0.478i)19-s + (−1.69 − 0.576i)22-s + (−0.130 − 0.991i)24-s + (−0.793 − 0.608i)25-s + ⋯
L(s)  = 1  + (0.130 + 0.991i)2-s + (0.965 + 0.258i)3-s + (−0.965 + 0.258i)4-s + (−0.130 + 0.991i)6-s + (−0.382 − 0.923i)8-s + (0.866 + 0.499i)9-s + (−0.793 + 1.60i)11-s − 12-s + (0.866 − 0.5i)16-s + (0.991 − 0.130i)17-s + (−0.382 + 0.923i)18-s + (−0.198 + 0.478i)19-s + (−1.69 − 0.576i)22-s + (−0.130 − 0.991i)24-s + (−0.793 − 0.608i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.452 - 0.891i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ -0.452 - 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.359925422\)
\(L(\frac12)\) \(\approx\) \(1.359925422\)
\(L(1)\) 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 \( 1 + (-0.965 - 0.258i)T \)
17 \( 1 + (-0.991 + 0.130i)T \)
good5 \( 1 + (0.793 + 0.608i)T^{2} \)
7 \( 1 + (-0.793 + 0.608i)T^{2} \)
11 \( 1 + (0.793 - 1.60i)T + (-0.608 - 0.793i)T^{2} \)
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 \( 1 + (0.583 - 0.665i)T + (-0.130 - 0.991i)T^{2} \)
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 \( 1 + (0.389 + 1.95i)T + (-0.923 + 0.382i)T^{2} \)
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 \( 1 + (1.24 + 1.42i)T + (-0.130 + 0.991i)T^{2} \)
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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

−9.993216371114324985136261275656, −9.281111510182323752297144252703, −8.390297197995631807464168225577, −7.58721552823969833503919156342, −7.29380339114055665302711023289, −6.03716476569327327476936513214, −4.99994738635407295287690673236, −4.30831525036670332145327229361, −3.31740949235383221384582629243, −2.02431511468338289870117393505, 1.15365313996020063893563516932, 2.54813662424279697355239889413, 3.23390929770440501890556000627, 4.06823919925416464029184682859, 5.31493613761198230804346447507, 6.13606559567737601747717628771, 7.60778141758776658251405888833, 8.168468580189094501193617859812, 8.958939725721921988419743863581, 9.609595008292483034667071801482

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