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Properties

Label 2-555-555.158-c1-0-46
Degree $2$
Conductor $555$
Sign $0.520 + 0.853i$
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  + (−2.45 + 0.658i)2-s + (1.51 − 0.842i)3-s + (3.87 − 2.23i)4-s + (2.23 − 0.0579i)5-s + (−3.16 + 3.06i)6-s + (−3.82 − 1.02i)7-s + (−4.44 + 4.44i)8-s + (1.57 − 2.55i)9-s + (−5.45 + 1.61i)10-s + 2.14i·11-s + (3.97 − 6.64i)12-s + (1.01 − 3.78i)13-s + 10.0·14-s + (3.33 − 1.97i)15-s + (3.52 − 6.10i)16-s + (−0.952 + 0.255i)17-s + ⋯
L(s)  = 1  + (−1.73 + 0.465i)2-s + (0.873 − 0.486i)3-s + (1.93 − 1.11i)4-s + (0.999 − 0.0259i)5-s + (−1.29 + 1.25i)6-s + (−1.44 − 0.387i)7-s + (−1.57 + 1.57i)8-s + (0.526 − 0.850i)9-s + (−1.72 + 0.510i)10-s + 0.645i·11-s + (1.14 − 1.91i)12-s + (0.281 − 1.04i)13-s + 2.69·14-s + (0.860 − 0.509i)15-s + (0.881 − 1.52i)16-s + (−0.231 + 0.0619i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.520 + 0.853i$
Analytic conductor: \(4.43169\)
Root analytic conductor: \(2.10515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 555,\ (\ :1/2),\ 0.520 + 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.758057 - 0.425611i\)
\(L(\frac12)\) \(\approx\) \(0.758057 - 0.425611i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.51 + 0.842i)T \)
5 \( 1 + (-2.23 + 0.0579i)T \)
37 \( 1 + (6.05 + 0.629i)T \)
good2 \( 1 + (2.45 - 0.658i)T + (1.73 - i)T^{2} \)
7 \( 1 + (3.82 + 1.02i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 - 2.14iT - 11T^{2} \)
13 \( 1 + (-1.01 + 3.78i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.952 - 0.255i)T + (14.7 - 8.5i)T^{2} \)
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 \( 1 + (3.13 + 11.7i)T + (-45.8 + 26.5i)T^{2} \)
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 \( 1 + (-5.36 - 1.43i)T + (58.0 + 33.5i)T^{2} \)
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 \( 1 + (-3.22 - 12.0i)T + (-71.8 + 41.5i)T^{2} \)
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

−10.12952226939798963587600650760, −9.369794365099557926792336384634, −9.252230964640863560206635645110, −8.009143517780686112647982943217, −7.12437036454911168223094139895, −6.65738129692934591743078899294, −5.61230564437468166809289996376, −3.28638301035923072875312077188, −2.21500055798314008599444210100, −0.807749147061707440248205802371, 1.60383988161725147072992793606, 2.74701957836712428189672620146, 3.55152137354044432386997049691, 5.67912672252079726434545871547, 6.74118467637713752581713557781, 7.63488072402494425376701550613, 8.836915292196246313671186601274, 9.329118158627433866494794310274, 9.636304283859582279475649528119, 10.48996701781411268471300174536

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