Properties

Label 2-855-1.1-c1-0-26
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·2-s − 1.82·4-s + 5-s + 1.41·7-s − 1.58·8-s + 0.414·10-s − 6.24·11-s − 0.585·13-s + 0.585·14-s + 3·16-s − 6.82·17-s − 19-s − 1.82·20-s − 2.58·22-s + 3.65·23-s + 25-s − 0.242·26-s − 2.58·28-s + 1.41·29-s − 8.82·31-s + 4.41·32-s − 2.82·34-s + 1.41·35-s − 0.585·37-s − 0.414·38-s − 1.58·40-s − 8.24·41-s + ⋯
L(s)  = 1  + 0.292·2-s − 0.914·4-s + 0.447·5-s + 0.534·7-s − 0.560·8-s + 0.130·10-s − 1.88·11-s − 0.162·13-s + 0.156·14-s + 0.750·16-s − 1.65·17-s − 0.229·19-s − 0.408·20-s − 0.551·22-s + 0.762·23-s + 0.200·25-s − 0.0475·26-s − 0.488·28-s + 0.262·29-s − 1.58·31-s + 0.780·32-s − 0.485·34-s + 0.239·35-s − 0.0963·37-s − 0.0671·38-s − 0.250·40-s − 1.28·41-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: \(1\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.414T + 2T^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 + 6.24T + 11T^{2} \)
13 \( 1 + 0.585T + 13T^{2} \)
17 \( 1 + 6.82T + 17T^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 8.82T + 31T^{2} \)
37 \( 1 + 0.585T + 37T^{2} \)
41 \( 1 + 8.24T + 41T^{2} \)
43 \( 1 - 3.75T + 43T^{2} \)
47 \( 1 + 3.65T + 47T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 - 4.48T + 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 - 1.65T + 67T^{2} \)
71 \( 1 - 5.17T + 71T^{2} \)
73 \( 1 - 3.65T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 7.17T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 18.2T + 97T^{2} \)
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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

−9.722745648379049918646211507621, −8.866871035962801030941946926719, −8.214413845173848281060607772219, −7.25258869154653640498590323148, −6.04254267696844185646324438513, −5.03561476211193453184203592506, −4.69800476778969242459409458966, −3.24134624149838719801586204400, −2.06707568249052481512345127651, 0, 2.06707568249052481512345127651, 3.24134624149838719801586204400, 4.69800476778969242459409458966, 5.03561476211193453184203592506, 6.04254267696844185646324438513, 7.25258869154653640498590323148, 8.214413845173848281060607772219, 8.866871035962801030941946926719, 9.722745648379049918646211507621

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