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Properties

Label 2-547-547.9-c1-0-18
Degree $2$
Conductor $547$
Sign $-0.907 - 0.419i$
Analytic cond. $4.36781$
Root an. cond. $2.08993$
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  + (−0.455 + 1.99i)2-s + (1.77 + 0.853i)3-s + (−1.96 − 0.946i)4-s + (2.51 + 3.15i)5-s + (−2.50 + 3.14i)6-s + (2.12 − 1.02i)7-s + (0.232 − 0.291i)8-s + (0.544 + 0.683i)9-s + (−7.42 + 3.57i)10-s + 0.713·11-s + (−2.67 − 3.35i)12-s + (−0.876 − 3.84i)13-s + (1.07 + 4.70i)14-s + (1.76 + 7.73i)15-s + (−2.24 − 2.81i)16-s + (0.909 + 1.14i)17-s + ⋯
L(s)  = 1  + (−0.321 + 1.40i)2-s + (1.02 + 0.493i)3-s + (−0.983 − 0.473i)4-s + (1.12 + 1.40i)5-s + (−1.02 + 1.28i)6-s + (0.803 − 0.387i)7-s + (0.0821 − 0.102i)8-s + (0.181 + 0.227i)9-s + (−2.34 + 1.13i)10-s + 0.214·11-s + (−0.773 − 0.969i)12-s + (−0.243 − 1.06i)13-s + (0.287 + 1.25i)14-s + (0.455 + 1.99i)15-s + (−0.561 − 0.704i)16-s + (0.220 + 0.276i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(547\)
Sign: $-0.907 - 0.419i$
Analytic conductor: \(4.36781\)
Root analytic conductor: \(2.08993\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 547,\ (\ :1/2),\ -0.907 - 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.448097 + 2.03581i\)
\(L(\frac12)\) \(\approx\) \(0.448097 + 2.03581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 + (17.8 - 15.1i)T \)
good2 \( 1 + (0.455 - 1.99i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (-1.77 - 0.853i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-2.51 - 3.15i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (-2.12 + 1.02i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 - 0.713T + 11T^{2} \)
13 \( 1 + (0.876 + 3.84i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-0.909 - 1.14i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + (-0.669 - 2.93i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (1.66 + 7.28i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-4.87 + 6.11i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-0.314 - 0.151i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (1.37 + 1.72i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 7.45T + 41T^{2} \)
43 \( 1 + (8.04 - 3.87i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 - 0.499T + 47T^{2} \)
53 \( 1 + (-0.402 - 1.76i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 - 2.30T + 59T^{2} \)
61 \( 1 + (7.09 + 8.89i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (9.30 + 11.6i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (1.40 + 1.75i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (0.546 - 2.39i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (-3.67 + 1.76i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + (9.22 - 11.5i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-1.36 - 6.00i)T + (-87.3 + 42.0i)T^{2} \)
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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

−10.61381512956795851513490954843, −10.12749181627128683978964927731, −9.297230556200416184758140084258, −8.170209455608313535643200888418, −7.83343948023841502226542039985, −6.59371506993546451809705112846, −6.05818801350088505744082674101, −4.85739358055541903435979365915, −3.32713842619139575962788044444, −2.30909228613956045869899182414, 1.53465111845206933070395927220, 1.81427934030595303522558161529, 3.05045592919256090821387606125, 4.55181716642554737587346562173, 5.45231276690524584893552668514, 6.98537299773513356855769111358, 8.417479017368412352972877631164, 8.826738540182002131064714147104, 9.435196159123101438507181077469, 10.20466218173976889804881551486

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