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Properties

Label 2-552-24.11-c1-0-67
Degree $2$
Conductor $552$
Sign $0.951 + 0.308i$
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  + (−0.218 + 1.39i)2-s + (0.262 + 1.71i)3-s + (−1.90 − 0.611i)4-s − 0.130·5-s + (−2.44 − 0.00827i)6-s − 3.77i·7-s + (1.27 − 2.52i)8-s + (−2.86 + 0.897i)9-s + (0.0285 − 0.182i)10-s − 4.36i·11-s + (0.547 − 3.42i)12-s − 5.50i·13-s + (5.27 + 0.826i)14-s + (−0.0342 − 0.223i)15-s + (3.25 + 2.32i)16-s + 0.402i·17-s + ⋯
L(s)  = 1  + (−0.154 + 0.987i)2-s + (0.151 + 0.988i)3-s + (−0.952 − 0.305i)4-s − 0.0583·5-s + (−0.999 − 0.00338i)6-s − 1.42i·7-s + (0.449 − 0.893i)8-s + (−0.954 + 0.299i)9-s + (0.00902 − 0.0576i)10-s − 1.31i·11-s + (0.158 − 0.987i)12-s − 1.52i·13-s + (1.41 + 0.220i)14-s + (−0.00883 − 0.0577i)15-s + (0.813 + 0.582i)16-s + 0.0975i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.951 + 0.308i$
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: \((2,\ 552,\ (\ :1/2),\ 0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821451 - 0.130036i\)
\(L(\frac12)\) \(\approx\) \(0.821451 - 0.130036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.218 - 1.39i)T \)
3 \( 1 + (-0.262 - 1.71i)T \)
23 \( 1 - T \)
good5 \( 1 + 0.130T + 5T^{2} \)
7 \( 1 + 3.77iT - 7T^{2} \)
11 \( 1 + 4.36iT - 11T^{2} \)
13 \( 1 + 5.50iT - 13T^{2} \)
17 \( 1 - 0.402iT - 17T^{2} \)
19 \( 1 + 2.08T + 19T^{2} \)
29 \( 1 + 8.77T + 29T^{2} \)
31 \( 1 - 7.75iT - 31T^{2} \)
37 \( 1 + 10.1iT - 37T^{2} \)
41 \( 1 - 2.80iT - 41T^{2} \)
43 \( 1 - 3.07T + 43T^{2} \)
47 \( 1 - 3.41T + 47T^{2} \)
53 \( 1 + 0.336T + 53T^{2} \)
59 \( 1 - 9.68iT - 59T^{2} \)
61 \( 1 + 2.78iT - 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 - 8.19T + 71T^{2} \)
73 \( 1 - 0.417T + 73T^{2} \)
79 \( 1 + 3.09iT - 79T^{2} \)
83 \( 1 - 3.59iT - 83T^{2} \)
89 \( 1 + 11.1iT - 89T^{2} \)
97 \( 1 - 4.53T + 97T^{2} \)
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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

−10.66199303306648061861493589653, −9.835453769119590755259199110799, −8.855076469541446940463200896622, −8.052077442578439055809508270320, −7.33452651233868926871605799340, −5.97339472638553280230884864280, −5.31067438077855226061706339202, −4.04437879173489846136653801701, −3.42258957449619005048755805274, −0.48952156571604758879591709904, 1.92536386284910978914731541491, 2.30612707626149084248724290467, 3.91855628062134273879718230733, 5.15717162721718454580734095782, 6.28031592621472733064317150973, 7.41607719906124417893626787650, 8.339696392617991072844565017926, 9.267115373428959727997763900754, 9.644835615910399603215083239681, 11.27045648211929599753347072593

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