Properties

Label 2-855-1.1-c1-0-5
Degree $2$
Conductor $855$
Sign $1$
Analytic cond. $6.82720$
Root an. cond. $2.61289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.48·2-s + 0.193·4-s + 5-s − 1.67·7-s + 2.67·8-s − 1.48·10-s − 0.481·11-s − 1.28·13-s + 2.48·14-s − 4.35·16-s + 6.15·17-s − 19-s + 0.193·20-s + 0.712·22-s − 2.54·23-s + 25-s + 1.90·26-s − 0.324·28-s + 9.83·29-s − 9.50·31-s + 1.09·32-s − 9.11·34-s − 1.67·35-s + 11.5·37-s + 1.48·38-s + 2.67·40-s − 5.18·41-s + ⋯
L(s)  = 1  − 1.04·2-s + 0.0969·4-s + 0.447·5-s − 0.633·7-s + 0.945·8-s − 0.468·10-s − 0.145·11-s − 0.357·13-s + 0.663·14-s − 1.08·16-s + 1.49·17-s − 0.229·19-s + 0.0433·20-s + 0.151·22-s − 0.530·23-s + 0.200·25-s + 0.373·26-s − 0.0613·28-s + 1.82·29-s − 1.70·31-s + 0.193·32-s − 1.56·34-s − 0.283·35-s + 1.90·37-s + 0.240·38-s + 0.422·40-s − 0.809·41-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.8033978008\)
\(L(\frac12)\) \(\approx\) \(0.8033978008\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 + 1.48T + 2T^{2} \)
7 \( 1 + 1.67T + 7T^{2} \)
11 \( 1 + 0.481T + 11T^{2} \)
13 \( 1 + 1.28T + 13T^{2} \)
17 \( 1 - 6.15T + 17T^{2} \)
23 \( 1 + 2.54T + 23T^{2} \)
29 \( 1 - 9.83T + 29T^{2} \)
31 \( 1 + 9.50T + 31T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 + 5.18T + 41T^{2} \)
43 \( 1 - 2.06T + 43T^{2} \)
47 \( 1 - 5.76T + 47T^{2} \)
53 \( 1 - 1.84T + 53T^{2} \)
59 \( 1 - 14.4T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 - 4.57T + 67T^{2} \)
71 \( 1 + 9.66T + 71T^{2} \)
73 \( 1 - 9.27T + 73T^{2} \)
79 \( 1 - 6.88T + 79T^{2} \)
83 \( 1 - 2.54T + 83T^{2} \)
89 \( 1 - 4.48T + 89T^{2} \)
97 \( 1 - 3.54T + 97T^{2} \)
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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

−9.972375953854413976359343338652, −9.511554413444072804518397829206, −8.565727048552894494883276661379, −7.81219046867850646273482745576, −6.96742659488225208381850525604, −5.91981490208671518682563033443, −4.93043375045715855466958776327, −3.67810498109511649627630025589, −2.32694895185668078901512999427, −0.861125345227534765918109386559, 0.861125345227534765918109386559, 2.32694895185668078901512999427, 3.67810498109511649627630025589, 4.93043375045715855466958776327, 5.91981490208671518682563033443, 6.96742659488225208381850525604, 7.81219046867850646273482745576, 8.565727048552894494883276661379, 9.511554413444072804518397829206, 9.972375953854413976359343338652

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