Properties

Label 2-552-184.83-c1-0-2
Degree $2$
Conductor $552$
Sign $-0.147 - 0.988i$
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  + (−1.41 − 0.0363i)2-s + (−0.654 − 0.755i)3-s + (1.99 + 0.102i)4-s + (0.0760 + 0.529i)5-s + (0.898 + 1.09i)6-s + (0.261 − 0.571i)7-s + (−2.82 − 0.218i)8-s + (−0.142 + 0.989i)9-s + (−0.0883 − 0.750i)10-s + (−0.877 + 2.98i)11-s + (−1.23 − 1.57i)12-s + (−2.47 + 1.13i)13-s + (−0.389 + 0.798i)14-s + (0.350 − 0.403i)15-s + (3.97 + 0.410i)16-s + (−1.18 + 1.83i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0257i)2-s + (−0.378 − 0.436i)3-s + (0.998 + 0.0514i)4-s + (0.0340 + 0.236i)5-s + (0.366 + 0.445i)6-s + (0.0986 − 0.216i)7-s + (−0.997 − 0.0770i)8-s + (−0.0474 + 0.329i)9-s + (−0.0279 − 0.237i)10-s + (−0.264 + 0.900i)11-s + (−0.355 − 0.455i)12-s + (−0.687 + 0.313i)13-s + (−0.104 + 0.213i)14-s + (0.0903 − 0.104i)15-s + (0.994 + 0.102i)16-s + (−0.286 + 0.445i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.147 - 0.988i$
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: \((2,\ 552,\ (\ :1/2),\ -0.147 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.301181 + 0.349595i\)
\(L(\frac12)\) \(\approx\) \(0.301181 + 0.349595i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0363i)T \)
3 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (4.54 + 1.54i)T \)
good5 \( 1 + (-0.0760 - 0.529i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (-0.261 + 0.571i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (0.877 - 2.98i)T + (-9.25 - 5.94i)T^{2} \)
13 \( 1 + (2.47 - 1.13i)T + (8.51 - 9.82i)T^{2} \)
17 \( 1 + (1.18 - 1.83i)T + (-7.06 - 15.4i)T^{2} \)
19 \( 1 + (-0.100 - 0.156i)T + (-7.89 + 17.2i)T^{2} \)
29 \( 1 + (3.62 - 5.63i)T + (-12.0 - 26.3i)T^{2} \)
31 \( 1 + (3.10 + 2.68i)T + (4.41 + 30.6i)T^{2} \)
37 \( 1 + (1.19 - 8.33i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-1.23 - 8.57i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-1.23 + 1.06i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 + 0.461iT - 47T^{2} \)
53 \( 1 + (0.0163 - 0.0358i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-4.75 - 10.4i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (8.18 - 9.44i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-0.973 - 3.31i)T + (-56.3 + 36.2i)T^{2} \)
71 \( 1 + (1.33 + 4.54i)T + (-59.7 + 38.3i)T^{2} \)
73 \( 1 + (2.73 - 1.75i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (0.176 + 0.386i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (4.29 + 0.616i)T + (79.6 + 23.3i)T^{2} \)
89 \( 1 + (-7.15 + 6.20i)T + (12.6 - 88.0i)T^{2} \)
97 \( 1 + (3.11 - 0.447i)T + (93.0 - 27.3i)T^{2} \)
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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

−10.79820055757316162462479535904, −10.22783563831784456745404091894, −9.335965822011312592201974809400, −8.302763287013919238441380797029, −7.38986734253020855381755397670, −6.83001095747467824726168853362, −5.79899315463522099637719323468, −4.46043039554648339490501113032, −2.73093861716541294665709692130, −1.58365184775039516267369233729, 0.37242350280074427776628638111, 2.25380716108610591003918465670, 3.60278100397533448981452093008, 5.20571765781184463143527114755, 5.92565189653181234996539035589, 7.10104828974695922314487696756, 7.999990469130981736544570243195, 8.943723698849656726164556558930, 9.574427137841377894005508648819, 10.55135304649461878752758401680

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