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Properties

Label 2-555-37.34-c1-0-24
Degree $2$
Conductor $555$
Sign $0.375 + 0.926i$
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  + (1.13 + 0.949i)2-s + (0.766 − 0.642i)3-s + (0.0319 + 0.181i)4-s + (−0.939 − 0.342i)5-s + 1.47·6-s + (−4.55 − 1.65i)7-s + (1.34 − 2.32i)8-s + (0.173 − 0.984i)9-s + (−0.738 − 1.27i)10-s + (0.708 − 1.22i)11-s + (0.140 + 0.118i)12-s + (−0.906 − 5.13i)13-s + (−3.58 − 6.20i)14-s + (−0.939 + 0.342i)15-s + (4.07 − 1.48i)16-s + (−0.403 + 2.28i)17-s + ⋯
L(s)  = 1  + (0.800 + 0.671i)2-s + (0.442 − 0.371i)3-s + (0.0159 + 0.0905i)4-s + (−0.420 − 0.152i)5-s + 0.603·6-s + (−1.72 − 0.627i)7-s + (0.474 − 0.821i)8-s + (0.0578 − 0.328i)9-s + (−0.233 − 0.404i)10-s + (0.213 − 0.369i)11-s + (0.0406 + 0.0341i)12-s + (−0.251 − 1.42i)13-s + (−0.958 − 1.65i)14-s + (−0.242 + 0.0883i)15-s + (1.01 − 0.370i)16-s + (−0.0978 + 0.554i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.375 + 0.926i$
Analytic conductor: \(4.43169\)
Root analytic conductor: \(2.10515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 555,\ (\ :1/2),\ 0.375 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49847 - 1.00920i\)
\(L(\frac12)\) \(\approx\) \(1.49847 - 1.00920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (0.180 - 6.08i)T \)
good2 \( 1 + (-1.13 - 0.949i)T + (0.347 + 1.96i)T^{2} \)
7 \( 1 + (4.55 + 1.65i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.708 + 1.22i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.906 + 5.13i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.403 - 2.28i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (-2.42 + 2.03i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (-4.67 - 8.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.267 + 0.463i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.0353T + 31T^{2} \)
41 \( 1 + (1.42 + 8.08i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 1.89T + 43T^{2} \)
47 \( 1 + (4.43 + 7.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.00 + 0.364i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-9.48 + 3.45i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.52 - 8.66i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-9.15 - 3.33i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-5.41 + 4.54i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + 9.34T + 73T^{2} \)
79 \( 1 + (13.6 + 4.95i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.880 + 4.99i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-15.0 + 5.47i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-6.69 - 11.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.35806307278642451676177477634, −9.821085507746482540241853275273, −8.764942705082090022324277788813, −7.49215675053321877174550845234, −7.00551319644049638194878691260, −6.05699741187725223118489698903, −5.16721266890102983612156232713, −3.65897237246971096182303427011, −3.24867087214638173133942214321, −0.77304444033703665844142395112, 2.37991582483890349419806138159, 3.15402003548516933655888287670, 4.05757581336377827714305961949, 4.95997694516276238047982694903, 6.38596661262122120465423374265, 7.19823359321659270095450285129, 8.532500032603183895359283260336, 9.322011342005387782897319496480, 10.02767839966940799876321626514, 11.18077405990820445282163739526

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