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Properties

Label 2-1224-1224.227-c0-0-1
Degree $2$
Conductor $1224$
Sign $0.0988 - 0.995i$
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 + (0.258 + 0.965i)3-s + (−0.258 + 0.965i)4-s + (0.608 − 0.793i)6-s + (0.923 − 0.382i)8-s + (−0.866 + 0.499i)9-s + (−0.991 + 1.13i)11-s − 12-s + (−0.866 − 0.499i)16-s + (−0.793 + 0.608i)17-s + (0.923 + 0.382i)18-s + (1.78 + 0.739i)19-s + (1.50 + 0.0983i)22-s + (0.608 + 0.793i)24-s + (−0.991 − 0.130i)25-s + ⋯
L(s)  = 1  + (−0.608 − 0.793i)2-s + (0.258 + 0.965i)3-s + (−0.258 + 0.965i)4-s + (0.608 − 0.793i)6-s + (0.923 − 0.382i)8-s + (−0.866 + 0.499i)9-s + (−0.991 + 1.13i)11-s − 12-s + (−0.866 − 0.499i)16-s + (−0.793 + 0.608i)17-s + (0.923 + 0.382i)18-s + (1.78 + 0.739i)19-s + (1.50 + 0.0983i)22-s + (0.608 + 0.793i)24-s + (−0.991 − 0.130i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0988 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0988 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.0988 - 0.995i$
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: \((2,\ 1224,\ (\ :0),\ 0.0988 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6502311107\)
\(L(\frac12)\) \(\approx\) \(0.6502311107\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.965i)T \)
17 \( 1 + (0.793 - 0.608i)T \)
good5 \( 1 + (0.991 + 0.130i)T^{2} \)
7 \( 1 + (-0.991 + 0.130i)T^{2} \)
11 \( 1 + (0.991 - 1.13i)T + (-0.130 - 0.991i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (-1.78 - 0.739i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (1.18 + 0.583i)T + (0.608 + 0.793i)T^{2} \)
43 \( 1 + (0.207 - 1.57i)T + (-0.965 - 0.258i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-1.57 - 1.20i)T + (0.258 + 0.965i)T^{2} \)
61 \( 1 + (0.991 - 0.130i)T^{2} \)
67 \( 1 + (1.05 - 0.608i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.357 - 0.534i)T + (-0.382 - 0.923i)T^{2} \)
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 \( 1 + (-1.78 + 0.882i)T + (0.608 - 0.793i)T^{2} \)
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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

−10.07097243652785925530141792529, −9.579294177865101850187172169695, −8.670900430119906578837668651949, −7.901588099438230766371814045061, −7.22349237719952297640260430665, −5.66285919304389007339760071740, −4.72842931092049577340168758635, −3.90344089852698011692020676622, −2.92497422688629586677484737947, −1.91552220416952761684477984650, 0.65921437882066926637795292547, 2.19051713456973382273844555207, 3.34885487829197579345790499674, 5.08852282598426764395067035417, 5.66452390123737120176360138102, 6.64265779607683889492590904884, 7.36821055443297705324178647918, 7.987546792678352006646708647112, 8.757452517249370961733443165113, 9.401706835323370489276622881637

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