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Properties

Label 2-552-23.13-c1-0-10
Degree $2$
Conductor $552$
Sign $0.608 + 0.793i$
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 + (0.645 − 1.41i)5-s + (−4.42 − 1.29i)7-s + (0.415 + 0.909i)9-s + (1.62 − 1.87i)11-s + (5.73 − 1.68i)13-s + (1.30 − 0.839i)15-s + (−0.0166 − 0.115i)17-s + (1.01 − 7.05i)19-s + (−3.02 − 3.48i)21-s + (3.84 − 2.87i)23-s + (1.69 + 1.95i)25-s + (−0.142 + 0.989i)27-s + (−0.381 − 2.65i)29-s + (−2.93 + 1.88i)31-s + ⋯
L(s)  = 1  + (0.485 + 0.312i)3-s + (0.288 − 0.631i)5-s + (−1.67 − 0.491i)7-s + (0.138 + 0.303i)9-s + (0.488 − 0.564i)11-s + (1.59 − 0.467i)13-s + (0.337 − 0.216i)15-s + (−0.00403 − 0.0280i)17-s + (0.232 − 1.61i)19-s + (−0.659 − 0.760i)21-s + (0.800 − 0.599i)23-s + (0.338 + 0.390i)25-s + (−0.0273 + 0.190i)27-s + (−0.0709 − 0.493i)29-s + (−0.526 + 0.338i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.608 + 0.793i$
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: \((2,\ 552,\ (\ :1/2),\ 0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39915 - 0.690340i\)
\(L(\frac12)\) \(\approx\) \(1.39915 - 0.690340i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.84 + 2.87i)T \)
good5 \( 1 + (-0.645 + 1.41i)T + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (4.42 + 1.29i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-1.62 + 1.87i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-5.73 + 1.68i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (0.0166 + 0.115i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (-1.01 + 7.05i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.381 + 2.65i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (2.93 - 1.88i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (1.92 + 4.21i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (4.80 - 10.5i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-4.37 - 2.81i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + (2.67 + 0.785i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (7.30 - 2.14i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (-8.77 + 5.64i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (-6.33 - 7.30i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (2.14 + 2.47i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-1.32 + 9.19i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-3.27 + 0.960i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-4.53 - 9.94i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-7.33 - 4.71i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (1.10 - 2.41i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.63507442192634812898120380797, −9.464133817645852671225143935274, −9.137871669537069816130037478376, −8.242837799909917477154857898141, −6.84826437812790757476655163786, −6.22336606377083475242721387562, −4.97188713512069433584443981327, −3.67142334673333742339572604024, −3.01234055169515105742783315147, −0.920354851785822265523017752086, 1.74601899151039839938115958280, 3.19896587364979204216962652074, 3.76204703722978820830495011294, 5.71067866206571885388798919684, 6.49214951101366451669315698610, 7.04596767108538585863002079270, 8.409718668898726649237801738427, 9.224112985101837997971613054863, 9.889157025983720246663475090399, 10.76566088896877768441988054271

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