Properties

Label 2-552-24.11-c1-0-27
Degree $2$
Conductor $552$
Sign $0.602 + 0.797i$
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  + (−0.484 − 1.32i)2-s + (−1.59 − 0.676i)3-s + (−1.53 + 1.28i)4-s + 1.67·5-s + (−0.126 + 2.44i)6-s − 2.28i·7-s + (2.45 + 1.41i)8-s + (2.08 + 2.15i)9-s + (−0.809 − 2.22i)10-s + 2.91i·11-s + (3.31 − 1.01i)12-s + 4.63i·13-s + (−3.03 + 1.10i)14-s + (−2.66 − 1.13i)15-s + (0.689 − 3.94i)16-s + 3.10i·17-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.920 − 0.390i)3-s + (−0.765 + 0.643i)4-s + 0.747·5-s + (−0.0516 + 0.998i)6-s − 0.864i·7-s + (0.866 + 0.499i)8-s + (0.695 + 0.718i)9-s + (−0.256 − 0.702i)10-s + 0.877i·11-s + (0.955 − 0.293i)12-s + 1.28i·13-s + (−0.812 + 0.295i)14-s + (−0.688 − 0.291i)15-s + (0.172 − 0.985i)16-s + 0.752i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.602 + 0.797i$
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: \((2,\ 552,\ (\ :1/2),\ 0.602 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.874168 - 0.435103i\)
\(L(\frac12)\) \(\approx\) \(0.874168 - 0.435103i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.484 + 1.32i)T \)
3 \( 1 + (1.59 + 0.676i)T \)
23 \( 1 + T \)
good5 \( 1 - 1.67T + 5T^{2} \)
7 \( 1 + 2.28iT - 7T^{2} \)
11 \( 1 - 2.91iT - 11T^{2} \)
13 \( 1 - 4.63iT - 13T^{2} \)
17 \( 1 - 3.10iT - 17T^{2} \)
19 \( 1 - 7.70T + 19T^{2} \)
29 \( 1 - 8.53T + 29T^{2} \)
31 \( 1 + 7.88iT - 31T^{2} \)
37 \( 1 + 3.97iT - 37T^{2} \)
41 \( 1 - 5.01iT - 41T^{2} \)
43 \( 1 - 7.62T + 43T^{2} \)
47 \( 1 + 4.61T + 47T^{2} \)
53 \( 1 - 0.427T + 53T^{2} \)
59 \( 1 - 9.10iT - 59T^{2} \)
61 \( 1 + 9.06iT - 61T^{2} \)
67 \( 1 + 8.29T + 67T^{2} \)
71 \( 1 - 4.33T + 71T^{2} \)
73 \( 1 - 6.91T + 73T^{2} \)
79 \( 1 + 6.46iT - 79T^{2} \)
83 \( 1 - 12.7iT - 83T^{2} \)
89 \( 1 + 4.52iT - 89T^{2} \)
97 \( 1 + 8.58T + 97T^{2} \)
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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

−10.63546986921643140533271217685, −9.897499641934170867333860096147, −9.391765652099841010819089460081, −7.87207661208308946360316201577, −7.15301690347573097472472347947, −6.06654418197863261361220091671, −4.82339519610856135285346526289, −4.01554857820772419319339299247, −2.19428478355843384188498935376, −1.16181856835889047404984239248, 0.931535538243228739983574378738, 3.17232213066464346939493469017, 4.94657479717583705165169478668, 5.54079307434036932823878270924, 6.08197273586085588710277126640, 7.16554832347776015131736645796, 8.296165600779959384550105999164, 9.221168385570490416423849901064, 9.943010158877751834895088863145, 10.61581058423505507994988298388

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