Properties

Label 2-552-184.85-c1-0-18
Degree $2$
Conductor $552$
Sign $0.363 + 0.931i$
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  + (−0.682 − 1.23i)2-s + (−0.540 − 0.841i)3-s + (−1.06 + 1.69i)4-s + (−3.63 + 1.66i)5-s + (−0.673 + 1.24i)6-s + (−2.20 + 0.646i)7-s + (2.82 + 0.170i)8-s + (−0.415 + 0.909i)9-s + (4.54 + 3.37i)10-s + (3.86 − 3.34i)11-s + (1.99 − 0.0146i)12-s + (−0.959 + 3.26i)13-s + (2.30 + 2.28i)14-s + (3.36 + 2.16i)15-s + (−1.71 − 3.61i)16-s + (0.278 − 1.93i)17-s + ⋯
L(s)  = 1  + (−0.482 − 0.875i)2-s + (−0.312 − 0.485i)3-s + (−0.534 + 0.845i)4-s + (−1.62 + 0.743i)5-s + (−0.274 + 0.507i)6-s + (−0.832 + 0.244i)7-s + (0.998 + 0.0603i)8-s + (−0.138 + 0.303i)9-s + (1.43 + 1.06i)10-s + (1.16 − 1.00i)11-s + (0.577 − 0.00423i)12-s + (−0.266 + 0.906i)13-s + (0.615 + 0.611i)14-s + (0.868 + 0.558i)15-s + (−0.428 − 0.903i)16-s + (0.0674 − 0.469i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.363 + 0.931i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.363 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.476806 - 0.325634i\)
\(L(\frac12)\) \(\approx\) \(0.476806 - 0.325634i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.682 + 1.23i)T \)
3 \( 1 + (0.540 + 0.841i)T \)
23 \( 1 + (-0.605 + 4.75i)T \)
good5 \( 1 + (3.63 - 1.66i)T + (3.27 - 3.77i)T^{2} \)
7 \( 1 + (2.20 - 0.646i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (-3.86 + 3.34i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (0.959 - 3.26i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.278 + 1.93i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (0.154 - 0.0222i)T + (18.2 - 5.35i)T^{2} \)
29 \( 1 + (-6.53 - 0.939i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (-2.58 - 1.66i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (3.20 + 1.46i)T + (24.2 + 27.9i)T^{2} \)
41 \( 1 + (0.499 + 1.09i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-6.20 - 9.66i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 - 4.85T + 47T^{2} \)
53 \( 1 + (0.766 + 2.61i)T + (-44.5 + 28.6i)T^{2} \)
59 \( 1 + (0.0453 - 0.154i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-8.37 + 13.0i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (-5.36 - 4.64i)T + (9.53 + 66.3i)T^{2} \)
71 \( 1 + (-7.52 + 8.68i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-1.97 - 13.7i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-1.49 - 0.439i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-2.31 - 1.05i)T + (54.3 + 62.7i)T^{2} \)
89 \( 1 + (-2.54 + 1.63i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (4.82 + 10.5i)T + (-63.5 + 73.3i)T^{2} \)
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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

−11.03718582617606169695756294747, −9.825779226404223256252568293058, −8.796631002628209651124827796964, −8.125239626148861182096894010150, −6.97549827070107617126930272677, −6.50810920768590990017189910932, −4.52248002213949933048341788854, −3.58855681684461159531792096550, −2.72426881717573231511060360266, −0.67307658760559274985081213694, 0.806210693098949818720715974549, 3.69433229656431497177912529361, 4.36678416785219363677995930355, 5.36646466436664883683323380763, 6.63246573543431836245961942154, 7.38569652433186402777170507231, 8.254236398380626706395681384512, 9.113915631214628914730895210467, 9.884260789196032092729740111377, 10.73533201676107383010524258517

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