Properties

Label 2-8550-1.1-c1-0-12
Degree $2$
Conductor $8550$
Sign $1$
Analytic cond. $68.2720$
Root an. cond. $8.26269$
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-s + 4-s − 3.16·7-s + 8-s − 4.16·11-s − 5.16·13-s − 3.16·14-s + 16-s + 0.837·17-s − 19-s − 4.16·22-s − 5.32·23-s − 5.16·26-s − 3.16·28-s + 3.83·29-s + 5.32·31-s + 32-s + 0.837·34-s − 10·37-s − 38-s + 6.32·41-s + 9.16·43-s − 4.16·44-s − 5.32·46-s + 6·47-s + 3.00·49-s − 5.16·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.19·7-s + 0.353·8-s − 1.25·11-s − 1.43·13-s − 0.845·14-s + 0.250·16-s + 0.203·17-s − 0.229·19-s − 0.887·22-s − 1.11·23-s − 1.01·26-s − 0.597·28-s + 0.712·29-s + 0.956·31-s + 0.176·32-s + 0.143·34-s − 1.64·37-s − 0.162·38-s + 0.987·41-s + 1.39·43-s − 0.627·44-s − 0.785·46-s + 0.875·47-s + 0.428·49-s − 0.715·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8550\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(68.2720\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.692039499\)
\(L(\frac12)\) \(\approx\) \(1.692039499\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 + T \)
good7 \( 1 + 3.16T + 7T^{2} \)
11 \( 1 + 4.16T + 11T^{2} \)
13 \( 1 + 5.16T + 13T^{2} \)
17 \( 1 - 0.837T + 17T^{2} \)
23 \( 1 + 5.32T + 23T^{2} \)
29 \( 1 - 3.83T + 29T^{2} \)
31 \( 1 - 5.32T + 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 - 6.32T + 41T^{2} \)
43 \( 1 - 9.16T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 1.83T + 53T^{2} \)
59 \( 1 - 7.48T + 59T^{2} \)
61 \( 1 + 4.16T + 61T^{2} \)
67 \( 1 - 6.48T + 67T^{2} \)
71 \( 1 + 1.16T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + 5.32T + 79T^{2} \)
83 \( 1 - 6.48T + 83T^{2} \)
89 \( 1 - 7.32T + 89T^{2} \)
97 \( 1 + 0.513T + 97T^{2} \)
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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

−7.57076358815131771695104923354, −7.08399089089709235337870191360, −6.28822373817863597007067618630, −5.69963725456638937402969940213, −4.99957294470904471274111249907, −4.30720961261910261123643747590, −3.43969800046570053904159901297, −2.65183866431982298611019665457, −2.22542675445859455050375977027, −0.52789055097457835436908702821, 0.52789055097457835436908702821, 2.22542675445859455050375977027, 2.65183866431982298611019665457, 3.43969800046570053904159901297, 4.30720961261910261123643747590, 4.99957294470904471274111249907, 5.69963725456638937402969940213, 6.28822373817863597007067618630, 7.08399089089709235337870191360, 7.57076358815131771695104923354

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