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Properties

Label 2-555-555.47-c1-0-71
Degree $2$
Conductor $555$
Sign $0.471 - 0.881i$
Analytic cond. $4.43169$
Root an. cond. $2.10515$
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  + (−0.658 − 2.45i)2-s + (−0.842 − 1.51i)3-s + (−3.87 + 2.23i)4-s + (1.16 − 1.90i)5-s + (−3.16 + 3.06i)6-s + (1.02 − 3.82i)7-s + (4.44 + 4.44i)8-s + (−1.57 + 2.55i)9-s + (−5.45 − 1.61i)10-s + 2.14i·11-s + (6.64 + 3.97i)12-s + (−3.78 − 1.01i)13-s − 10.0·14-s + (−3.86 − 0.159i)15-s + (3.52 − 6.10i)16-s + (−0.255 − 0.952i)17-s + ⋯
L(s)  = 1  + (−0.465 − 1.73i)2-s + (−0.486 − 0.873i)3-s + (−1.93 + 1.11i)4-s + (0.522 − 0.852i)5-s + (−1.29 + 1.25i)6-s + (0.387 − 1.44i)7-s + (1.57 + 1.57i)8-s + (−0.526 + 0.850i)9-s + (−1.72 − 0.510i)10-s + 0.645i·11-s + (1.91 + 1.14i)12-s + (−1.04 − 0.281i)13-s − 2.69·14-s + (−0.999 − 0.0412i)15-s + (0.881 − 1.52i)16-s + (−0.0619 − 0.231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(4.43169\)
Root analytic conductor: \(2.10515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 555,\ (\ :1/2),\ 0.471 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.504226 + 0.302008i\)
\(L(\frac12)\) \(\approx\) \(0.504226 + 0.302008i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.842 + 1.51i)T \)
5 \( 1 + (-1.16 + 1.90i)T \)
37 \( 1 + (-0.629 + 6.05i)T \)
good2 \( 1 + (0.658 + 2.45i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (-1.02 + 3.82i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 - 2.14iT - 11T^{2} \)
13 \( 1 + (3.78 + 1.01i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.255 + 0.952i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.86 - 3.38i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.15 + 2.15i)T + 23iT^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 - 2.81T + 31T^{2} \)
41 \( 1 + (-5.90 + 3.41i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.705 - 0.705i)T - 43iT^{2} \)
47 \( 1 + (4.99 - 4.99i)T - 47iT^{2} \)
53 \( 1 + (11.7 - 3.13i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.10 + 3.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.30 + 5.72i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.43 - 5.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-5.69 + 3.29i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.32 + 1.32i)T - 73iT^{2} \)
79 \( 1 + (6.83 - 3.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.0 + 3.22i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-2.22 + 3.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.28 + 4.28i)T + 97iT^{2} \)
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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

−10.33639621849023977955026153253, −9.548701227695951842918805468186, −8.317730420092260035990265524427, −7.74873761340985454514335575071, −6.45346018741341791584611057487, −4.81729390105642971653327917880, −4.30909902202198993413161899409, −2.48263921862912018521586220591, −1.52789419316912125224364991708, −0.42916663703104155712527595518, 2.70212549673923502493253694545, 4.59641712510424059458403066769, 5.30480886619176824094231804529, 6.28182169994354301476875280883, 6.55896035328146999954572186935, 8.073412305806523907790977459722, 8.772081751556087735291833743055, 9.538465450188596489150705728913, 10.24468457035446813710480214029, 11.28365268631070279741550981422

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