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Properties

Label 2-1224-1224.635-c0-0-0
Degree $2$
Conductor $1224$
Sign $0.860 + 0.510i$
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 + (−0.130 − 0.991i)3-s + (0.965 − 0.258i)4-s + (0.258 + 0.965i)6-s + (−0.923 + 0.382i)8-s + (−0.965 + 0.258i)9-s + (0.483 + 1.42i)11-s + (−0.382 − 0.923i)12-s + (0.866 − 0.5i)16-s + (0.793 − 0.608i)17-s + (0.923 − 0.382i)18-s + (1.78 + 0.739i)19-s + (−0.665 − 1.34i)22-s + (0.5 + 0.866i)24-s + (0.608 − 0.793i)25-s + ⋯
L(s)  = 1  + (−0.991 + 0.130i)2-s + (−0.130 − 0.991i)3-s + (0.965 − 0.258i)4-s + (0.258 + 0.965i)6-s + (−0.923 + 0.382i)8-s + (−0.965 + 0.258i)9-s + (0.483 + 1.42i)11-s + (−0.382 − 0.923i)12-s + (0.866 − 0.5i)16-s + (0.793 − 0.608i)17-s + (0.923 − 0.382i)18-s + (1.78 + 0.739i)19-s + (−0.665 − 1.34i)22-s + (0.5 + 0.866i)24-s + (0.608 − 0.793i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.860 + 0.510i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (635, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.860 + 0.510i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6945362654\)
\(L(\frac12)\) \(\approx\) \(0.6945362654\)
\(L(1)\) 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 \( 1 + (0.130 + 0.991i)T \)
17 \( 1 + (-0.793 + 0.608i)T \)
good5 \( 1 + (-0.608 + 0.793i)T^{2} \)
7 \( 1 + (0.608 + 0.793i)T^{2} \)
11 \( 1 + (-0.483 - 1.42i)T + (-0.793 + 0.608i)T^{2} \)
13 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (-1.78 - 0.739i)T + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.130 - 0.991i)T^{2} \)
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 \( 1 + (0.0862 + 1.31i)T + (-0.991 + 0.130i)T^{2} \)
43 \( 1 + (1.25 + 0.965i)T + (0.258 + 0.965i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.258 - 1.96i)T + (-0.965 - 0.258i)T^{2} \)
61 \( 1 + (-0.608 - 0.793i)T^{2} \)
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 \( 1 + (-0.793 + 0.608i)T^{2} \)
83 \( 1 + (0.241 + 1.83i)T + (-0.965 + 0.258i)T^{2} \)
89 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
97 \( 1 + (0.130 - 1.99i)T + (-0.991 - 0.130i)T^{2} \)
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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

−9.841529483824647146403514953817, −8.981914367879709427526252420292, −8.123282520218122371363629730388, −7.21325971855908445838193340531, −7.06911652452816275532806011791, −5.88839491248595182541829502376, −5.08231230465159549013910130188, −3.33242417609314373030617251008, −2.17152424372232444348032018282, −1.15996574714772711073154016525, 1.14324621730396476211768511027, 3.12412028817409094678243498653, 3.40877859366528601316659288432, 4.99911548401625355251149007327, 5.87438749443541813800071464626, 6.67958326422457024954174384182, 7.903326544991633703828470810115, 8.471899841151600872058260648293, 9.461700223832334693985962656074, 9.682881191988666803143382715684

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