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Properties

Label 2-855-95.79-c0-0-1
Degree $2$
Conductor $855$
Sign $-0.643 - 0.765i$
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  + (1.50 + 1.26i)2-s + (0.500 + 2.83i)4-s + (−0.984 − 0.173i)5-s + (−1.85 + 3.20i)8-s + (−1.26 − 1.50i)10-s + (−4.14 + 1.50i)16-s + (0.984 − 1.17i)17-s + (0.939 − 0.342i)19-s − 2.87i·20-s + (−0.984 + 0.173i)23-s + (0.939 + 0.342i)25-s + (0.592 − 0.342i)31-s + (−4.68 − 1.70i)32-s + (2.97 − 0.524i)34-s + (1.85 + 0.673i)38-s + ⋯
L(s)  = 1  + (1.50 + 1.26i)2-s + (0.500 + 2.83i)4-s + (−0.984 − 0.173i)5-s + (−1.85 + 3.20i)8-s + (−1.26 − 1.50i)10-s + (−4.14 + 1.50i)16-s + (0.984 − 1.17i)17-s + (0.939 − 0.342i)19-s − 2.87i·20-s + (−0.984 + 0.173i)23-s + (0.939 + 0.342i)25-s + (0.592 − 0.342i)31-s + (−4.68 − 1.70i)32-s + (2.97 − 0.524i)34-s + (1.85 + 0.673i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.643 - 0.765i$
Analytic conductor: \(0.426700\)
Root analytic conductor: \(0.653223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :0),\ -0.643 - 0.765i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.822637441\)
\(L(\frac12)\) \(\approx\) \(1.822637441\)
\(L(1)\) 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 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
good2 \( 1 + (-1.50 - 1.26i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.984 + 1.17i)T + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.984 - 0.173i)T + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.223 - 0.266i)T + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.223 + 1.26i)T + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (1.62 - 0.939i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.173 + 0.984i)T^{2} \)
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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

−11.25971397964623721537468786390, −9.621039032503213709979442021212, −8.498617820416133846352956548012, −7.71111649074451413248582129315, −7.30141270478841283952329701110, −6.28125969121373127675582699430, −5.29886011031915661593898930936, −4.63318983818943320003405896266, −3.65404427421823926443778220702, −2.87784850081031620399614930302, 1.38831248782282148853153857985, 2.84621437658689094270792164105, 3.67904891440590125885950879255, 4.34280093148485401582782682600, 5.42659061116101975336640654292, 6.20189115865413823234911742148, 7.30876654625183756757320847956, 8.451756277087487083132382280779, 9.811010524725768854217118096551, 10.35099772568244578812785006554

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