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Properties

Label 2-855-285.239-c0-0-3
Degree $2$
Conductor $855$
Sign $-0.140 + 0.990i$
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  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.965 − 0.258i)5-s − i·7-s + (0.366 − 1.36i)10-s + 1.41i·11-s + (−0.866 + 0.5i)13-s + (−1.22 − 0.707i)14-s + (0.499 − 0.866i)16-s + (−0.707 + 1.22i)17-s − 19-s + (−0.707 − 0.707i)20-s + (1.73 + 1.00i)22-s + (0.866 − 0.499i)25-s + 1.41i·26-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.965 − 0.258i)5-s − i·7-s + (0.366 − 1.36i)10-s + 1.41i·11-s + (−0.866 + 0.5i)13-s + (−1.22 − 0.707i)14-s + (0.499 − 0.866i)16-s + (−0.707 + 1.22i)17-s − 19-s + (−0.707 − 0.707i)20-s + (1.73 + 1.00i)22-s + (0.866 − 0.499i)25-s + 1.41i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.140 + 0.990i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ -0.140 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.579939632\)
\(L(\frac12)\) \(\approx\) \(1.579939632\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.965 + 0.258i)T \)
19 \( 1 + T \)
good2 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
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

−10.41177566570129050448484918430, −9.728275409660440819544167683001, −8.791332305348813197809227863255, −7.39126551200379994747363372762, −6.71565256721627415342410500985, −5.37848726164791307807736040613, −4.45619847879320397552493108914, −3.91864714273835124298356460726, −2.26785905066122544984885625244, −1.75626674969022821957048800529, 2.21657494564902857814218173734, 3.31801497022179435851217584209, 4.99558899609758938512559458928, 5.32759696654638446185878901185, 6.33355056353826803531097088680, 6.74437307680150392318108071609, 7.999246631694452734502622285862, 8.726902043627315316638965535864, 9.581483974816391732470634614460, 10.60041420870134055844448022340

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