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Properties

Label 2-1224-1224.1163-c0-0-0
Degree $2$
Conductor $1224$
Sign $-0.787 - 0.616i$
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  + (−0.793 + 0.608i)2-s + (−0.608 − 0.793i)3-s + (0.258 − 0.965i)4-s + (0.965 + 0.258i)6-s + (0.382 + 0.923i)8-s + (−0.258 + 0.965i)9-s + (−1.78 + 0.117i)11-s + (−0.923 + 0.382i)12-s + (−0.866 − 0.499i)16-s + (−0.991 + 0.130i)17-s + (−0.382 − 0.923i)18-s + (−0.198 + 0.478i)19-s + (1.34 − 1.18i)22-s + (0.499 − 0.866i)24-s + (−0.130 + 0.991i)25-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)2-s + (−0.608 − 0.793i)3-s + (0.258 − 0.965i)4-s + (0.965 + 0.258i)6-s + (0.382 + 0.923i)8-s + (−0.258 + 0.965i)9-s + (−1.78 + 0.117i)11-s + (−0.923 + 0.382i)12-s + (−0.866 − 0.499i)16-s + (−0.991 + 0.130i)17-s + (−0.382 − 0.923i)18-s + (−0.198 + 0.478i)19-s + (1.34 − 1.18i)22-s + (0.499 − 0.866i)24-s + (−0.130 + 0.991i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.787 - 0.616i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (1163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ -0.787 - 0.616i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1611230558\)
\(L(\frac12)\) \(\approx\) \(0.1611230558\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.793 - 0.608i)T \)
3 \( 1 + (0.608 + 0.793i)T \)
17 \( 1 + (0.991 - 0.130i)T \)
good5 \( 1 + (0.130 - 0.991i)T^{2} \)
7 \( 1 + (-0.130 - 0.991i)T^{2} \)
11 \( 1 + (1.78 - 0.117i)T + (0.991 - 0.130i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.198 - 0.478i)T + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.608 - 0.793i)T^{2} \)
29 \( 1 + (-0.793 - 0.608i)T^{2} \)
31 \( 1 + (0.991 + 0.130i)T^{2} \)
37 \( 1 + (0.382 + 0.923i)T^{2} \)
41 \( 1 + (-0.284 - 0.837i)T + (-0.793 + 0.608i)T^{2} \)
43 \( 1 + (1.96 + 0.258i)T + (0.965 + 0.258i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.965 - 1.25i)T + (-0.258 - 0.965i)T^{2} \)
61 \( 1 + (0.130 + 0.991i)T^{2} \)
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 \( 1 + (0.991 - 0.130i)T^{2} \)
83 \( 1 + (-0.465 - 0.607i)T + (-0.258 + 0.965i)T^{2} \)
89 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
97 \( 1 + (0.608 - 1.79i)T + (-0.793 - 0.608i)T^{2} \)
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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

−10.37749327227673819354170967034, −9.346994600442752814590859072939, −8.298487176866833662039858460542, −7.80995423995414182426478972059, −7.05884633965612593695569310153, −6.22545198558075233919040515378, −5.43266304796735416143786150689, −4.72296221069421538125918850053, −2.69691609445899603045608824293, −1.64374155176984921668211168363, 0.18406933482493440676209624857, 2.26933004679280757940272604713, 3.23070378410460942576856887700, 4.39828840634726528686066600015, 5.17069424902255962174164657820, 6.34049584117945890045729847463, 7.23674254077701001531701703713, 8.309119796771092040543070715549, 8.824343628163623520019849179603, 9.974388484000598467937234937670

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