Properties

Label 2-855-1.1-c1-0-24
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  + 2.64·2-s + 5.00·4-s − 5-s + 1.64·7-s + 7.93·8-s − 2.64·10-s − 0.354·11-s − 0.354·13-s + 4.35·14-s + 11.0·16-s + 4·17-s − 19-s − 5.00·20-s − 0.937·22-s − 9.29·23-s + 25-s − 0.937·26-s + 8.22·28-s − 8.93·29-s + 6·31-s + 13.2·32-s + 10.5·34-s − 1.64·35-s + 3.64·37-s − 2.64·38-s − 7.93·40-s + 9.64·41-s + ⋯
L(s)  = 1  + 1.87·2-s + 2.50·4-s − 0.447·5-s + 0.622·7-s + 2.80·8-s − 0.836·10-s − 0.106·11-s − 0.0982·13-s + 1.16·14-s + 2.75·16-s + 0.970·17-s − 0.229·19-s − 1.11·20-s − 0.199·22-s − 1.93·23-s + 0.200·25-s − 0.183·26-s + 1.55·28-s − 1.65·29-s + 1.07·31-s + 2.33·32-s + 1.81·34-s − 0.278·35-s + 0.599·37-s − 0.429·38-s − 1.25·40-s + 1.50·41-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

\(L(1)\) \(\approx\) \(4.719630185\)
\(L(\frac12)\) \(\approx\) \(4.719630185\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64T + 2T^{2} \)
7 \( 1 - 1.64T + 7T^{2} \)
11 \( 1 + 0.354T + 11T^{2} \)
13 \( 1 + 0.354T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
23 \( 1 + 9.29T + 23T^{2} \)
29 \( 1 + 8.93T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 3.64T + 37T^{2} \)
41 \( 1 - 9.64T + 41T^{2} \)
43 \( 1 - 5.64T + 43T^{2} \)
47 \( 1 - 1.29T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + 6.58T + 67T^{2} \)
71 \( 1 + 7.29T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 6.58T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 + 2.93T + 97T^{2} \)
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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

−10.67271511362287841677187630475, −9.516934784813309500688880154964, −7.80089205681565877940778521648, −7.71036418310382896784177414998, −6.27725230823798878770474023582, −5.71537394554387100462234331872, −4.64581092554730096159220447587, −4.02563579301204193992383587525, −2.98971781948011649321538550987, −1.79827696998905858967763090168, 1.79827696998905858967763090168, 2.98971781948011649321538550987, 4.02563579301204193992383587525, 4.64581092554730096159220447587, 5.71537394554387100462234331872, 6.27725230823798878770474023582, 7.71036418310382896784177414998, 7.80089205681565877940778521648, 9.516934784813309500688880154964, 10.67271511362287841677187630475

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