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Properties

Label 2-1224-1224.131-c0-0-0
Degree $2$
Conductor $1224$
Sign $0.519 + 0.854i$
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 + (−0.258 + 0.965i)3-s + (0.258 + 0.965i)4-s + (0.793 − 0.608i)6-s + (0.382 − 0.923i)8-s + (−0.866 − 0.499i)9-s + (0.130 − 1.99i)11-s − 12-s + (−0.866 + 0.499i)16-s + (0.608 − 0.793i)17-s + (0.382 + 0.923i)18-s + (−0.739 − 1.78i)19-s + (−1.31 + 1.50i)22-s + (0.793 + 0.608i)24-s + (0.130 + 0.991i)25-s + ⋯
L(s)  = 1  + (−0.793 − 0.608i)2-s + (−0.258 + 0.965i)3-s + (0.258 + 0.965i)4-s + (0.793 − 0.608i)6-s + (0.382 − 0.923i)8-s + (−0.866 − 0.499i)9-s + (0.130 − 1.99i)11-s − 12-s + (−0.866 + 0.499i)16-s + (0.608 − 0.793i)17-s + (0.382 + 0.923i)18-s + (−0.739 − 1.78i)19-s + (−1.31 + 1.50i)22-s + (0.793 + 0.608i)24-s + (0.130 + 0.991i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.519 + 0.854i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.519 + 0.854i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5850661369\)
\(L(\frac12)\) \(\approx\) \(0.5850661369\)
\(L(1)\) 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 \( 1 + (0.258 - 0.965i)T \)
17 \( 1 + (-0.608 + 0.793i)T \)
good5 \( 1 + (-0.130 - 0.991i)T^{2} \)
7 \( 1 + (0.130 - 0.991i)T^{2} \)
11 \( 1 + (-0.130 + 1.99i)T + (-0.991 - 0.130i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.739 + 1.78i)T + (-0.707 + 0.707i)T^{2} \)
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 \( 1 + (0.123 + 0.0420i)T + (0.793 + 0.608i)T^{2} \)
43 \( 1 + (-1.20 + 0.158i)T + (0.965 - 0.258i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.158 - 0.207i)T + (-0.258 + 0.965i)T^{2} \)
61 \( 1 + (-0.130 + 0.991i)T^{2} \)
67 \( 1 + (1.37 + 0.793i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (-1.75 - 0.349i)T + (0.923 + 0.382i)T^{2} \)
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 \( 1 + (-1.24 + 0.423i)T + (0.793 - 0.608i)T^{2} \)
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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

−9.723384023989680024113817482546, −8.991617458585994624735180079193, −8.649001758794502596711021093386, −7.54708331441526801927040417590, −6.47498372244054340268223160062, −5.53018599058001492768527252569, −4.44806076539492502434247673746, −3.36830375381062301667398026276, −2.77043201786802800445864158538, −0.71887054482917499697416772431, 1.51451134250683618344070732203, 2.25947148810411947617800555863, 4.19091302384168719117615546933, 5.31943355032621685272453283795, 6.19798306069803738151799888810, 6.81093337384586261316512472614, 7.72564771996770004207450348299, 8.095897080413019180660261817827, 9.114441263964195375839905941950, 10.18245204663297273267694100341

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