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Properties

Label 2-1224-1224.1091-c0-0-1
Degree $2$
Conductor $1224$
Sign $0.962 - 0.270i$
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  + (0.991 + 0.130i)2-s + (−0.965 + 0.258i)3-s + (0.965 + 0.258i)4-s + (−0.991 + 0.130i)6-s + (0.923 + 0.382i)8-s + (0.866 − 0.499i)9-s + (0.608 − 1.79i)11-s − 12-s + (0.866 + 0.5i)16-s + (−0.130 + 0.991i)17-s + (0.923 − 0.382i)18-s + (−0.478 + 0.198i)19-s + (0.837 − 1.69i)22-s + (−0.991 − 0.130i)24-s + (0.608 + 0.793i)25-s + ⋯
L(s)  = 1  + (0.991 + 0.130i)2-s + (−0.965 + 0.258i)3-s + (0.965 + 0.258i)4-s + (−0.991 + 0.130i)6-s + (0.923 + 0.382i)8-s + (0.866 − 0.499i)9-s + (0.608 − 1.79i)11-s − 12-s + (0.866 + 0.5i)16-s + (−0.130 + 0.991i)17-s + (0.923 − 0.382i)18-s + (−0.478 + 0.198i)19-s + (0.837 − 1.69i)22-s + (−0.991 − 0.130i)24-s + (0.608 + 0.793i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.962 - 0.270i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.962 - 0.270i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.546087156\)
\(L(\frac12)\) \(\approx\) \(1.546087156\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.991 - 0.130i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
17 \( 1 + (0.130 - 0.991i)T \)
good5 \( 1 + (-0.608 - 0.793i)T^{2} \)
7 \( 1 + (0.608 - 0.793i)T^{2} \)
11 \( 1 + (-0.608 + 1.79i)T + (-0.793 - 0.608i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.478 - 0.198i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.130 + 0.991i)T^{2} \)
29 \( 1 + (-0.991 + 0.130i)T^{2} \)
31 \( 1 + (-0.793 + 0.608i)T^{2} \)
37 \( 1 + (-0.923 - 0.382i)T^{2} \)
41 \( 1 + (-0.0420 + 0.641i)T + (-0.991 - 0.130i)T^{2} \)
43 \( 1 + (0.207 - 0.158i)T + (0.258 - 0.965i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.158 + 1.20i)T + (-0.965 + 0.258i)T^{2} \)
61 \( 1 + (-0.608 + 0.793i)T^{2} \)
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 \( 1 + (-0.793 - 0.608i)T^{2} \)
83 \( 1 + (-0.241 + 1.83i)T + (-0.965 - 0.258i)T^{2} \)
89 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
97 \( 1 + (-0.0578 - 0.882i)T + (-0.991 + 0.130i)T^{2} \)
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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

−10.42345897607214265504345726122, −9.118538683545096740140903098553, −8.238565904853663872041350453325, −7.15005111348707717375846010322, −6.17237264174717910908926374695, −5.94070161489935379706025809200, −4.90144818429414368756941163994, −3.94652153212352855586643070178, −3.21237040301529212478333312325, −1.46173903762931975090410616146, 1.53048921870367899154005131807, 2.61373437018127900333482910917, 4.22539587549111291412158357488, 4.67712624386051521246637929397, 5.54164560499517713920513041729, 6.72201252413816593269391576763, 6.88291727723288492073785143521, 7.87386109407122635398850980869, 9.384441534964708532956511788093, 10.13755041083736803637471286166

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