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Properties

Label 2-552-24.11-c1-0-0
Degree $2$
Conductor $552$
Sign $-0.772 - 0.634i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.29 + 0.569i)2-s + (0.611 − 1.62i)3-s + (1.35 − 1.47i)4-s − 2.61·5-s + (0.131 + 2.44i)6-s + 0.280i·7-s + (−0.907 + 2.67i)8-s + (−2.25 − 1.98i)9-s + (3.38 − 1.49i)10-s + 1.67i·11-s + (−1.56 − 3.09i)12-s + 0.357i·13-s + (−0.160 − 0.363i)14-s + (−1.60 + 4.24i)15-s + (−0.352 − 3.98i)16-s − 0.581i·17-s + ⋯
L(s)  = 1  + (−0.915 + 0.402i)2-s + (0.353 − 0.935i)3-s + (0.675 − 0.737i)4-s − 1.17·5-s + (0.0537 + 0.998i)6-s + 0.106i·7-s + (−0.320 + 0.947i)8-s + (−0.750 − 0.660i)9-s + (1.07 − 0.471i)10-s + 0.505i·11-s + (−0.451 − 0.892i)12-s + 0.0991i·13-s + (−0.0427 − 0.0972i)14-s + (−0.413 + 1.09i)15-s + (−0.0880 − 0.996i)16-s − 0.140i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.772 - 0.634i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.772 - 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0339864 + 0.0949320i\)
\(L(\frac12)\) \(\approx\) \(0.0339864 + 0.0949320i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 - 0.569i)T \)
3 \( 1 + (-0.611 + 1.62i)T \)
23 \( 1 + T \)
good5 \( 1 + 2.61T + 5T^{2} \)
7 \( 1 - 0.280iT - 7T^{2} \)
11 \( 1 - 1.67iT - 11T^{2} \)
13 \( 1 - 0.357iT - 13T^{2} \)
17 \( 1 + 0.581iT - 17T^{2} \)
19 \( 1 + 6.26T + 19T^{2} \)
29 \( 1 + 1.43T + 29T^{2} \)
31 \( 1 - 5.89iT - 31T^{2} \)
37 \( 1 - 7.11iT - 37T^{2} \)
41 \( 1 - 9.79iT - 41T^{2} \)
43 \( 1 + 4.96T + 43T^{2} \)
47 \( 1 - 6.01T + 47T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 + 2.28iT - 59T^{2} \)
61 \( 1 + 0.378iT - 61T^{2} \)
67 \( 1 + 8.97T + 67T^{2} \)
71 \( 1 + 6.12T + 71T^{2} \)
73 \( 1 + 1.10T + 73T^{2} \)
79 \( 1 + 15.7iT - 79T^{2} \)
83 \( 1 + 7.51iT - 83T^{2} \)
89 \( 1 + 5.80iT - 89T^{2} \)
97 \( 1 + 18.9T + 97T^{2} \)
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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

−11.18056796279912725232308787654, −10.16154120686350147372032956215, −9.007119816369243166041418516100, −8.333775397277315896446712577175, −7.67229546442552457517120698959, −6.89882000377219000202265436222, −6.10591300701560014812556145179, −4.58190563052875413198131022684, −3.03043199684872170152053099659, −1.65965930400900080808573332748, 0.07190342069686102082074347442, 2.39681519166619092338039155779, 3.68923450533276522455894755604, 4.20638734592595098375403341637, 5.87044401204867541887720386946, 7.24730896607644458919888965601, 8.100776759379650415340366123826, 8.673133548965757369006852599024, 9.511211603057112709760635405547, 10.61804047533677194941302158040

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