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Properties

Label 2-552-23.13-c1-0-4
Degree $2$
Conductor $552$
Sign $0.861 - 0.507i$
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  + (−0.841 − 0.540i)3-s + (−0.721 + 1.57i)5-s + (−2.41 − 0.709i)7-s + (0.415 + 0.909i)9-s + (2.10 − 2.43i)11-s + (5.18 − 1.52i)13-s + (1.46 − 0.938i)15-s + (0.538 + 3.74i)17-s + (−0.848 + 5.90i)19-s + (1.64 + 1.90i)21-s + (2.38 + 4.16i)23-s + (1.30 + 1.50i)25-s + (0.142 − 0.989i)27-s + (1.51 + 10.5i)29-s + (9.18 − 5.90i)31-s + ⋯
L(s)  = 1  + (−0.485 − 0.312i)3-s + (−0.322 + 0.706i)5-s + (−0.913 − 0.268i)7-s + (0.138 + 0.303i)9-s + (0.634 − 0.732i)11-s + (1.43 − 0.422i)13-s + (0.377 − 0.242i)15-s + (0.130 + 0.908i)17-s + (−0.194 + 1.35i)19-s + (0.359 + 0.415i)21-s + (0.497 + 0.867i)23-s + (0.260 + 0.300i)25-s + (0.0273 − 0.190i)27-s + (0.280 + 1.95i)29-s + (1.64 − 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.861 - 0.507i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.861 - 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06752 + 0.291162i\)
\(L(\frac12)\) \(\approx\) \(1.06752 + 0.291162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (-2.38 - 4.16i)T \)
good5 \( 1 + (0.721 - 1.57i)T + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (2.41 + 0.709i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-2.10 + 2.43i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-5.18 + 1.52i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.538 - 3.74i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (0.848 - 5.90i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-1.51 - 10.5i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-9.18 + 5.90i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (1.63 + 3.58i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-1.53 + 3.35i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (6.71 + 4.31i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 - 4.95T + 47T^{2} \)
53 \( 1 + (-10.3 - 3.03i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (4.59 - 1.35i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (6.18 - 3.97i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (6.79 + 7.84i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (-8.00 - 9.23i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (0.247 - 1.71i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-6.78 + 1.99i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (1.73 + 3.78i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-4.91 - 3.15i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-5.94 + 13.0i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.71244358555178499953798496650, −10.38078042155695359864340566469, −9.029562940478726652615588252058, −8.160540589515538296014904620818, −7.06615528805907650547248060433, −6.28573630080512079560751420589, −5.67469871308400611990803528151, −3.81947122094986224378009170058, −3.29230367002259883570932223197, −1.24059484700743669762482590164, 0.841896256148733182956508299848, 2.86404383288812119292672682996, 4.25566875660723686082149280934, 4.87658165670753548297921690963, 6.37827569344780959638085011731, 6.69498537062917792088397015482, 8.266863891183038073332495228118, 9.064444185711557098936137211989, 9.713712923028259610020225307211, 10.73004901352627233223061390014

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