Properties

Label 2-1224-1224.419-c0-0-1
Degree $2$
Conductor $1224$
Sign $0.838 - 0.544i$
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 + (1.24 − 0.423i)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 + (1.18 − 0.583i)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 + (1.24 − 0.423i)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 + (1.18 − 0.583i)22-s + (0.5 + 0.866i)24-s + (−0.608 + 0.793i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.838 - 0.544i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.838 - 0.544i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.007807514\)
\(L(\frac12)\) \(\approx\) \(2.007807514\)
\(L(1)\) 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 \( 1 + (-1.24 + 0.423i)T + (0.793 - 0.608i)T^{2} \)
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 \( 1 + (-1.50 + 0.0983i)T + (0.991 - 0.130i)T^{2} \)
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 \( 1 + (1.57 + 1.05i)T + (0.382 + 0.923i)T^{2} \)
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 \( 1 + (-0.130 - 0.00855i)T + (0.991 + 0.130i)T^{2} \)
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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

−10.26570928214288092219450048955, −9.111688876922513650795179574829, −8.656533936875713891654615741936, −7.34857998577127559776121147628, −6.30345749177852690716959207972, −5.80567055211901517020448968073, −4.47867029548697079139962247001, −4.15509496773469096406926599755, −3.14050757326945326487550271382, −1.96214863401798239993717702003, 1.68873102546366555531317735430, 2.53179617526972211247145990739, 3.83676730353034122871704501253, 4.59175167398295220337417372045, 5.92010852753724758540624409456, 6.48246365486326889639212447112, 7.07491683827259980883275317079, 8.063493045917027443104086942020, 8.759897735693884519056334030822, 9.894260875815049395855754641282

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