Properties

Label 2-552-24.11-c1-0-48
Degree $2$
Conductor $552$
Sign $0.615 + 0.787i$
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  + (−1.28 + 0.595i)2-s + (−1.03 + 1.38i)3-s + (1.29 − 1.52i)4-s + 1.31·5-s + (0.501 − 2.39i)6-s − 1.36i·7-s + (−0.744 + 2.72i)8-s + (−0.855 − 2.87i)9-s + (−1.69 + 0.786i)10-s + 1.33i·11-s + (0.785 + 3.37i)12-s − 4.58i·13-s + (0.813 + 1.75i)14-s + (−1.36 + 1.83i)15-s + (−0.670 − 3.94i)16-s − 4.76i·17-s + ⋯
L(s)  = 1  + (−0.906 + 0.421i)2-s + (−0.597 + 0.801i)3-s + (0.645 − 0.764i)4-s + 0.590·5-s + (0.204 − 0.978i)6-s − 0.516i·7-s + (−0.263 + 0.964i)8-s + (−0.285 − 0.958i)9-s + (−0.535 + 0.248i)10-s + 0.403i·11-s + (0.226 + 0.973i)12-s − 1.27i·13-s + (0.217 + 0.468i)14-s + (−0.352 + 0.472i)15-s + (−0.167 − 0.985i)16-s − 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.615 + 0.787i$
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: \((2,\ 552,\ (\ :1/2),\ 0.615 + 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.520193 - 0.253591i\)
\(L(\frac12)\) \(\approx\) \(0.520193 - 0.253591i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 0.595i)T \)
3 \( 1 + (1.03 - 1.38i)T \)
23 \( 1 + T \)
good5 \( 1 - 1.31T + 5T^{2} \)
7 \( 1 + 1.36iT - 7T^{2} \)
11 \( 1 - 1.33iT - 11T^{2} \)
13 \( 1 + 4.58iT - 13T^{2} \)
17 \( 1 + 4.76iT - 17T^{2} \)
19 \( 1 + 6.45T + 19T^{2} \)
29 \( 1 + 4.15T + 29T^{2} \)
31 \( 1 + 8.95iT - 31T^{2} \)
37 \( 1 - 2.96iT - 37T^{2} \)
41 \( 1 + 6.40iT - 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 + 4.02T + 47T^{2} \)
53 \( 1 - 9.40T + 53T^{2} \)
59 \( 1 - 4.77iT - 59T^{2} \)
61 \( 1 + 13.3iT - 61T^{2} \)
67 \( 1 - 7.21T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 - 3.85T + 73T^{2} \)
79 \( 1 - 0.671iT - 79T^{2} \)
83 \( 1 - 6.46iT - 83T^{2} \)
89 \( 1 - 4.72iT - 89T^{2} \)
97 \( 1 + 5.63T + 97T^{2} \)
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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

−10.49427909965174145901619726268, −9.786426914606545147533918860376, −9.177866695537224402087238456074, −8.017283481290428942947976383586, −7.07054936302062539451397030384, −6.00445839341455205347345525919, −5.38396960338118411255537842731, −4.12588007143926384286666985789, −2.41403524465434971132900336808, −0.47061362666224218587507171097, 1.59106185910355983350217452823, 2.37475867885945440955963028746, 4.12546070250741615252840591371, 5.82080290173815757600298539179, 6.40377689567843571121661002308, 7.34340637478427095759196710334, 8.480710272972796114951493597573, 8.983310767058892970432322681958, 10.19132544621734080419302351782, 10.89947005829213208774653488565

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