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Properties

Label 2-855-855.664-c0-0-1
Degree $2$
Conductor $855$
Sign $-0.766 - 0.642i$
Analytic cond. $0.426700$
Root an. cond. $0.653223$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 1.5i)2-s + (0.866 + 0.5i)3-s + (−1 − 1.73i)4-s + (−0.5 − 0.866i)5-s + (−1.5 + 0.866i)6-s + 1.73·8-s + (0.499 + 0.866i)9-s + 1.73·10-s + (−0.5 + 0.866i)11-s − 2i·12-s + (0.866 + 1.5i)13-s − 0.999i·15-s + (−0.5 + 0.866i)16-s − 1.73·18-s + 19-s + (−1 + 1.73i)20-s + ⋯
L(s)  = 1  + (−0.866 + 1.5i)2-s + (0.866 + 0.5i)3-s + (−1 − 1.73i)4-s + (−0.5 − 0.866i)5-s + (−1.5 + 0.866i)6-s + 1.73·8-s + (0.499 + 0.866i)9-s + 1.73·10-s + (−0.5 + 0.866i)11-s − 2i·12-s + (0.866 + 1.5i)13-s − 0.999i·15-s + (−0.5 + 0.866i)16-s − 1.73·18-s + 19-s + (−1 + 1.73i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (664, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ -0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7359500981\)
\(L(\frac12)\) \(\approx\) \(0.7359500981\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 - T \)
good2 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.73T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - 1.73T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−10.09222467588937018300594205308, −9.508592690290796994954704846555, −8.772779348527561609489020457190, −8.341103828584133095036947970202, −7.41425146114312093061131564608, −6.84827230937564762032108628279, −5.42306478138217979768920099003, −4.73067410441555718517988763018, −3.72386367125946137837225731320, −1.70010922306513803985162488895, 1.00250825715784666862581313098, 2.54076552451066672949282718003, 3.23180598214025908620397587490, 3.76425249592904197621231455632, 5.71468390341589072292670867176, 7.07599366160019243149766085552, 7.995459848054394942267555107011, 8.370848561997883860097468151779, 9.263120735784220000734079263803, 10.37062539857946704545770606774

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