Properties

Label 2-552-23.9-c1-0-9
Degree $2$
Conductor $552$
Sign $-0.635 + 0.772i$
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  + (−0.654 − 0.755i)3-s + (−0.412 − 2.86i)5-s + (1.09 − 2.39i)7-s + (−0.142 + 0.989i)9-s + (4.14 + 1.21i)11-s + (0.129 + 0.284i)13-s + (−1.89 + 2.18i)15-s + (−5.58 − 3.59i)17-s + (−4.72 + 3.03i)19-s + (−2.52 + 0.740i)21-s + (2.39 − 4.15i)23-s + (−3.25 + 0.954i)25-s + (0.841 − 0.540i)27-s + (−7.53 − 4.84i)29-s + (−0.650 + 0.751i)31-s + ⋯
L(s)  = 1  + (−0.378 − 0.436i)3-s + (−0.184 − 1.28i)5-s + (0.412 − 0.903i)7-s + (−0.0474 + 0.329i)9-s + (1.25 + 0.367i)11-s + (0.0360 + 0.0788i)13-s + (−0.489 + 0.565i)15-s + (−1.35 − 0.870i)17-s + (−1.08 + 0.696i)19-s + (−0.550 + 0.161i)21-s + (0.498 − 0.866i)23-s + (−0.650 + 0.190i)25-s + (0.161 − 0.104i)27-s + (−1.39 − 0.899i)29-s + (−0.116 + 0.134i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.635 + 0.772i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.635 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.478953 - 1.01400i\)
\(L(\frac12)\) \(\approx\) \(0.478953 - 1.01400i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (-2.39 + 4.15i)T \)
good5 \( 1 + (0.412 + 2.86i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (-1.09 + 2.39i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-4.14 - 1.21i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-0.129 - 0.284i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (5.58 + 3.59i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (4.72 - 3.03i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (7.53 + 4.84i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (0.650 - 0.751i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-0.640 + 4.45i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (0.241 + 1.68i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-6.05 - 6.98i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + (0.186 - 0.407i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (4.56 + 10.0i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (8.24 - 9.51i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (5.67 - 1.66i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (-4.55 + 1.33i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-2.71 + 1.74i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (1.36 + 2.98i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (1.39 - 9.69i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-3.70 - 4.27i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (1.46 + 10.1i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.70712807929573089181774155911, −9.331616781396528180318796524819, −8.811453795427090095895278841433, −7.72211312285027046466898962841, −6.89994138178022161574162934504, −5.89710582156483718645508892614, −4.47699369329184876903397684526, −4.24467513791711480969964313266, −1.94479586047614450284501001571, −0.68938783855956423362510405420, 2.07691715885743926762844195608, 3.40455789806161098146819809147, 4.38778574410698716127627759705, 5.75299035037891358749530951129, 6.48249058211160051554286585144, 7.30489534902413677105582979431, 8.805635893268007077365127192632, 9.121350362614140898851610731105, 10.58344848251519667307419837156, 11.04967010038704753545621704516

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