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Properties

Label 2-552-69.68-c1-0-10
Degree $2$
Conductor $552$
Sign $0.419 - 0.907i$
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  + (1.10 + 1.33i)3-s + 3.03·5-s + 2.60i·7-s + (−0.569 + 2.94i)9-s + 4.32·11-s − 6.76·13-s + (3.34 + 4.05i)15-s − 3.03·17-s − 5.50i·19-s + (−3.47 + 2.87i)21-s + (4.32 − 2.07i)23-s + 4.21·25-s + (−4.56 + 2.48i)27-s − 1.89i·29-s + 1.79·31-s + ⋯
L(s)  = 1  + (0.636 + 0.771i)3-s + 1.35·5-s + 0.984i·7-s + (−0.189 + 0.981i)9-s + 1.30·11-s − 1.87·13-s + (0.864 + 1.04i)15-s − 0.736·17-s − 1.26i·19-s + (−0.759 + 0.626i)21-s + (0.901 − 0.433i)23-s + 0.843·25-s + (−0.878 + 0.478i)27-s − 0.351i·29-s + 0.322·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.419 - 0.907i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.419 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81126 + 1.15805i\)
\(L(\frac12)\) \(\approx\) \(1.81126 + 1.15805i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.10 - 1.33i)T \)
23 \( 1 + (-4.32 + 2.07i)T \)
good5 \( 1 - 3.03T + 5T^{2} \)
7 \( 1 - 2.60iT - 7T^{2} \)
11 \( 1 - 4.32T + 11T^{2} \)
13 \( 1 + 6.76T + 13T^{2} \)
17 \( 1 + 3.03T + 17T^{2} \)
19 \( 1 + 5.50iT - 19T^{2} \)
29 \( 1 + 1.89iT - 29T^{2} \)
31 \( 1 - 1.79T + 31T^{2} \)
37 \( 1 + 3.16iT - 37T^{2} \)
41 \( 1 + 7.23iT - 41T^{2} \)
43 \( 1 - 2.93iT - 43T^{2} \)
47 \( 1 - 9.23iT - 47T^{2} \)
53 \( 1 + 1.55T + 53T^{2} \)
59 \( 1 + 5.06iT - 59T^{2} \)
61 \( 1 - 13.6iT - 61T^{2} \)
67 \( 1 + 8.41iT - 67T^{2} \)
71 \( 1 - 9.10iT - 71T^{2} \)
73 \( 1 - 0.729T + 73T^{2} \)
79 \( 1 + 14.4iT - 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 - 6.29T + 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−10.74538498179780519877244800299, −9.750059347473680140292834117915, −9.197595410642590351881769518941, −8.853581695718246691063767732009, −7.31166662745382874057128596682, −6.29759573770714202368268008445, −5.21614583889970112026851287211, −4.48984752845523630023062252068, −2.76327880060513217857903161352, −2.15040302867546888489592198551, 1.32992934454986032771565915747, 2.35322125469497166030226397059, 3.72106790994888352155631595986, 5.05311005958452554527051385388, 6.38394947144032745738389147861, 6.91138603768669043938111084210, 7.82341100096016379157966393855, 9.054919953026114931546879494273, 9.653308922577186768241702241139, 10.33055636187791300306854458180

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