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Properties

Label 2-1224-1224.1163-c0-0-1
Degree $2$
Conductor $1224$
Sign $-0.799 + 0.600i$
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 + 0.00855i)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 + (−0.0983 + 0.0862i)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 + 0.00855i)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 + (−0.0983 + 0.0862i)22-s + (−0.793 + 0.608i)24-s + (−0.130 + 0.991i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.799 + 0.600i$
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: \((2,\ 1224,\ (\ :0),\ -0.799 + 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.377818417\)
\(L(\frac12)\) \(\approx\) \(1.377818417\)
\(L(1)\) 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 \( 1 + (0.130 - 0.00855i)T + (0.991 - 0.130i)T^{2} \)
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 \( 1 + (-0.641 - 1.88i)T + (-0.793 + 0.608i)T^{2} \)
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 \( 1 + (-0.172 - 0.867i)T + (-0.923 + 0.382i)T^{2} \)
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 \( 1 + (-0.483 + 1.42i)T + (-0.793 - 0.608i)T^{2} \)
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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

−9.573900902523695655279055642962, −9.006373595023130665459128103271, −7.66123051050567096142917060475, −6.99052261580848479424097031208, −6.21167216715123662339427508200, −5.26278749530041115031297042763, −4.58315547595011132842251489831, −3.09685233921635067651156622970, −2.38009938440552065579192631013, −1.00152084646150806020137246015, 2.37902102240987527898923865686, 3.71199087370349229399879384128, 4.11135514211265523492821005367, 5.29576747528694254052097766918, 5.83584069496538501738805423860, 6.68457071120512943196600546417, 7.83311952434708810077001216471, 8.491270054446225389610114080588, 9.385198382528317794856490490812, 10.37906502369460126993070685868

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