Properties

Label 2-552-184.83-c1-0-27
Degree $2$
Conductor $552$
Sign $0.661 - 0.750i$
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  + (1.40 − 0.125i)2-s + (0.654 + 0.755i)3-s + (1.96 − 0.353i)4-s + (0.329 + 2.29i)5-s + (1.01 + 0.982i)6-s + (0.102 − 0.223i)7-s + (2.72 − 0.745i)8-s + (−0.142 + 0.989i)9-s + (0.752 + 3.18i)10-s + (−1.07 + 3.67i)11-s + (1.55 + 1.25i)12-s + (−4.96 + 2.26i)13-s + (0.115 − 0.327i)14-s + (−1.51 + 1.75i)15-s + (3.74 − 1.39i)16-s + (3.16 − 4.92i)17-s + ⋯
L(s)  = 1  + (0.996 − 0.0888i)2-s + (0.378 + 0.436i)3-s + (0.984 − 0.176i)4-s + (0.147 + 1.02i)5-s + (0.415 + 0.401i)6-s + (0.0385 − 0.0844i)7-s + (0.964 − 0.263i)8-s + (−0.0474 + 0.329i)9-s + (0.237 + 1.00i)10-s + (−0.325 + 1.10i)11-s + (0.449 + 0.362i)12-s + (−1.37 + 0.629i)13-s + (0.0309 − 0.0875i)14-s + (−0.391 + 0.451i)15-s + (0.937 − 0.348i)16-s + (0.767 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.661 - 0.750i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.661 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.72723 + 1.23205i\)
\(L(\frac12)\) \(\approx\) \(2.72723 + 1.23205i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.125i)T \)
3 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (0.134 + 4.79i)T \)
good5 \( 1 + (-0.329 - 2.29i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (-0.102 + 0.223i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (1.07 - 3.67i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (4.96 - 2.26i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (-3.16 + 4.92i)T + (-7.06 - 15.4i)T^{2} \)
19 \( 1 + (2.56 + 3.98i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (-5.43 + 8.46i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (-2.84 - 2.46i)T + (4.41 + 30.6i)T^{2} \)
37 \( 1 + (0.0873 - 0.607i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-1.08 - 7.51i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (4.07 - 3.53i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 + 7.48iT - 47T^{2} \)
53 \( 1 + (3.61 - 7.92i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (2.19 + 4.80i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-2.00 + 2.31i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-0.858 - 2.92i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (4.30 + 14.6i)T + (-59.7 + 38.3i)T^{2} \)
73 \( 1 + (-8.82 + 5.66i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (2.27 + 4.97i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-0.0564 - 0.00811i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (12.4 - 10.8i)T + (12.6 - 88.0i)T^{2} \)
97 \( 1 + (15.6 - 2.25i)T + (93.0 - 27.3i)T^{2} \)
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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

−10.90305146544705442649949268252, −10.08076167391339115928736091582, −9.592481081074585218849918895104, −7.912806424735727416804603580320, −7.07317445783832643000662715483, −6.46298564859873542394910450903, −4.87468007323700907965891103728, −4.48888021641195756802268171450, −2.83926467716461638985737581774, −2.42357257526597509649429714387, 1.42917136162576944233632625817, 2.84640043318234574599144842002, 3.92190488889674635284809803133, 5.26881731134041401160721675948, 5.71683002346990935385090753087, 6.98572506377086776834247582447, 8.064544852312630560286710848333, 8.505641354107571912051074970290, 9.923769846851734058863831827006, 10.75382073653253392171656710211

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