Properties

Label 2-855-5.4-c1-0-26
Degree $2$
Conductor $855$
Sign $-0.871 + 0.490i$
Analytic cond. $6.82720$
Root an. cond. $2.61289$
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  − 1.82i·2-s − 1.32·4-s + (−1.94 + 1.09i)5-s + 1.45i·7-s − 1.23i·8-s + (1.99 + 3.55i)10-s + 3.89·11-s − 3.05i·13-s + 2.64·14-s − 4.89·16-s − 3.92i·17-s + 19-s + (2.57 − 1.45i)20-s − 7.10i·22-s − 5.37i·23-s + ⋯
L(s)  = 1  − 1.28i·2-s − 0.660·4-s + (−0.871 + 0.490i)5-s + 0.548i·7-s − 0.437i·8-s + (0.632 + 1.12i)10-s + 1.17·11-s − 0.848i·13-s + 0.706·14-s − 1.22·16-s − 0.951i·17-s + 0.229·19-s + (0.575 − 0.324i)20-s − 1.51i·22-s − 1.12i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $-0.871 + 0.490i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -0.871 + 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.314200 - 1.19806i\)
\(L(\frac12)\) \(\approx\) \(0.314200 - 1.19806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.94 - 1.09i)T \)
19 \( 1 - T \)
good2 \( 1 + 1.82iT - 2T^{2} \)
7 \( 1 - 1.45iT - 7T^{2} \)
11 \( 1 - 3.89T + 11T^{2} \)
13 \( 1 + 3.05iT - 13T^{2} \)
17 \( 1 + 3.92iT - 17T^{2} \)
23 \( 1 + 5.37iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8.43T + 31T^{2} \)
37 \( 1 + 5.95iT - 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 1.45iT - 43T^{2} \)
47 \( 1 + 4.90iT - 47T^{2} \)
53 \( 1 + 4.23iT - 53T^{2} \)
59 \( 1 - 3.35T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 9.84iT - 67T^{2} \)
71 \( 1 + 8.64T + 71T^{2} \)
73 \( 1 - 2.43iT - 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 - 12.6iT - 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 3.05iT - 97T^{2} \)
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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

−10.07307769870945615413088265411, −9.151772883828519452436077111494, −8.396343213704546878265162724776, −7.18609442448184611324665025435, −6.52469126659464888191114483553, −5.10482415866519286995933906796, −3.94943107913009062109312888964, −3.22589175694575022491115138366, −2.25204804917240107294739616917, −0.64216144947282649167307904214, 1.50544756630966692402952392224, 3.61866432742605536984680181024, 4.37151454589885872487660082684, 5.36387945295630377213066763438, 6.47132997318405382993749358085, 7.06697180398603449679906295198, 7.83364885924227132322840395432, 8.673695333196501433742134946850, 9.244839906639423850939284890628, 10.50040913609446747705890575016

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