Properties

Label 2-552-184.91-c1-0-42
Degree $2$
Conductor $552$
Sign $0.320 + 0.947i$
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  + (0.869 − 1.11i)2-s + 3-s + (−0.489 − 1.93i)4-s + 1.54·5-s + (0.869 − 1.11i)6-s + 2.92·7-s + (−2.58 − 1.13i)8-s + 9-s + (1.34 − 1.72i)10-s + 4.46i·11-s + (−0.489 − 1.93i)12-s − 2.71i·13-s + (2.54 − 3.26i)14-s + 1.54·15-s + (−3.52 + 1.89i)16-s + 0.363i·17-s + ⋯
L(s)  = 1  + (0.614 − 0.788i)2-s + 0.577·3-s + (−0.244 − 0.969i)4-s + 0.689·5-s + (0.354 − 0.455i)6-s + 1.10·7-s + (−0.915 − 0.402i)8-s + 0.333·9-s + (0.423 − 0.544i)10-s + 1.34i·11-s + (−0.141 − 0.559i)12-s − 0.753i·13-s + (0.679 − 0.871i)14-s + 0.398·15-s + (−0.880 + 0.474i)16-s + 0.0881i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.320 + 0.947i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.320 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22291 - 1.59411i\)
\(L(\frac12)\) \(\approx\) \(2.22291 - 1.59411i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.869 + 1.11i)T \)
3 \( 1 - T \)
23 \( 1 + (3.23 + 3.53i)T \)
good5 \( 1 - 1.54T + 5T^{2} \)
7 \( 1 - 2.92T + 7T^{2} \)
11 \( 1 - 4.46iT - 11T^{2} \)
13 \( 1 + 2.71iT - 13T^{2} \)
17 \( 1 - 0.363iT - 17T^{2} \)
19 \( 1 - 1.20iT - 19T^{2} \)
29 \( 1 - 3.26iT - 29T^{2} \)
31 \( 1 + 1.19iT - 31T^{2} \)
37 \( 1 + 9.28T + 37T^{2} \)
41 \( 1 - 3.65T + 41T^{2} \)
43 \( 1 + 8.22iT - 43T^{2} \)
47 \( 1 - 2.10iT - 47T^{2} \)
53 \( 1 + 8.60T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 - 5.63T + 61T^{2} \)
67 \( 1 - 10.2iT - 67T^{2} \)
71 \( 1 + 6.80iT - 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 + 2.37T + 79T^{2} \)
83 \( 1 - 16.3iT - 83T^{2} \)
89 \( 1 - 2.99iT - 89T^{2} \)
97 \( 1 + 13.6iT - 97T^{2} \)
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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

−10.40191656832000943604750395691, −10.07099662572379673556260738772, −9.030096853997600449799167825892, −8.074707856574592298271125517539, −6.95032469533617347410660107374, −5.65055172707328310794888402833, −4.84116295908183989914353482491, −3.85596003978354399255823582785, −2.38518622034993666324356585077, −1.64495080816462194576567352857, 1.94498337781871337253826785995, 3.32852037599377670246220696523, 4.43606481958159079179840191093, 5.47316726877326157581967466105, 6.27543302699876661656015250166, 7.41397050979353331000988032033, 8.264166083614254019644669020181, 8.870027337087790399897525243489, 9.855277113888804304780012167125, 11.24062162436232628768070538675

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