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Properties

Label 2-552-24.11-c1-0-42
Degree $2$
Conductor $552$
Sign $0.506 + 0.862i$
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  + (−1.30 − 0.538i)2-s + (−1.71 + 0.226i)3-s + (1.42 + 1.40i)4-s + 3.49·5-s + (2.36 + 0.628i)6-s − 1.83i·7-s + (−1.10 − 2.60i)8-s + (2.89 − 0.776i)9-s + (−4.57 − 1.88i)10-s + 1.48i·11-s + (−2.75 − 2.09i)12-s − 6.77i·13-s + (−0.987 + 2.40i)14-s + (−6.00 + 0.791i)15-s + (0.0382 + 3.99i)16-s + 4.55i·17-s + ⋯
L(s)  = 1  + (−0.924 − 0.380i)2-s + (−0.991 + 0.130i)3-s + (0.710 + 0.703i)4-s + 1.56·5-s + (0.966 + 0.256i)6-s − 0.693i·7-s + (−0.389 − 0.921i)8-s + (0.965 − 0.258i)9-s + (−1.44 − 0.595i)10-s + 0.447i·11-s + (−0.796 − 0.604i)12-s − 1.87i·13-s + (−0.263 + 0.641i)14-s + (−1.55 + 0.204i)15-s + (0.00957 + 0.999i)16-s + 1.10i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.506 + 0.862i$
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: \((2,\ 552,\ (\ :1/2),\ 0.506 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.786491 - 0.450299i\)
\(L(\frac12)\) \(\approx\) \(0.786491 - 0.450299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 + 0.538i)T \)
3 \( 1 + (1.71 - 0.226i)T \)
23 \( 1 - T \)
good5 \( 1 - 3.49T + 5T^{2} \)
7 \( 1 + 1.83iT - 7T^{2} \)
11 \( 1 - 1.48iT - 11T^{2} \)
13 \( 1 + 6.77iT - 13T^{2} \)
17 \( 1 - 4.55iT - 17T^{2} \)
19 \( 1 - 0.397T + 19T^{2} \)
29 \( 1 - 0.0571T + 29T^{2} \)
31 \( 1 + 3.03iT - 31T^{2} \)
37 \( 1 + 9.08iT - 37T^{2} \)
41 \( 1 + 3.30iT - 41T^{2} \)
43 \( 1 + 7.31T + 43T^{2} \)
47 \( 1 - 9.07T + 47T^{2} \)
53 \( 1 + 1.34T + 53T^{2} \)
59 \( 1 + 4.89iT - 59T^{2} \)
61 \( 1 - 6.29iT - 61T^{2} \)
67 \( 1 - 7.37T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 + 6.29iT - 79T^{2} \)
83 \( 1 + 12.3iT - 83T^{2} \)
89 \( 1 - 16.4iT - 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−10.45530713193386974639613287592, −10.07367363604142427641333185079, −9.240129900215150710944286291245, −7.971648132058602577980286644047, −7.01905068275235129415422030976, −6.06635422408615886555167023842, −5.34954661903903642174036225597, −3.76387361584725287203964340925, −2.17199082870241622804372124627, −0.888603561442998008192446862823, 1.39566797575154306565806465497, 2.41866518227457246159078143923, 4.91197624020090251768139835775, 5.62083124495042482809861134979, 6.53469838425918512275590010550, 6.92225328703178055548339654114, 8.488391341671493400173427489734, 9.474006635085311241318716257682, 9.721687060614491723581842870284, 10.86453507814674571829235064501

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