Properties

Label 2-8550-1.1-c1-0-108
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2.73·7-s − 8-s − 0.267·11-s − 0.732·13-s − 2.73·14-s + 16-s + 4.19·17-s − 19-s + 0.267·22-s − 7.92·23-s + 0.732·26-s + 2.73·28-s − 1.73·29-s + 4.46·31-s − 32-s − 4.19·34-s + 2·37-s + 38-s − 10.9·41-s − 2.19·43-s − 0.267·44-s + 7.92·46-s − 3.46·47-s + 0.464·49-s − 0.732·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.03·7-s − 0.353·8-s − 0.0807·11-s − 0.203·13-s − 0.730·14-s + 0.250·16-s + 1.01·17-s − 0.229·19-s + 0.0571·22-s − 1.65·23-s + 0.143·26-s + 0.516·28-s − 0.321·29-s + 0.801·31-s − 0.176·32-s − 0.719·34-s + 0.328·37-s + 0.162·38-s − 1.70·41-s − 0.334·43-s − 0.0403·44-s + 1.16·46-s − 0.505·47-s + 0.0663·49-s − 0.101·52-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: \(1\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 + T \)
good7 \( 1 - 2.73T + 7T^{2} \)
11 \( 1 + 0.267T + 11T^{2} \)
13 \( 1 + 0.732T + 13T^{2} \)
17 \( 1 - 4.19T + 17T^{2} \)
23 \( 1 + 7.92T + 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
31 \( 1 - 4.46T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 2.19T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 1.73T + 53T^{2} \)
59 \( 1 + 2.19T + 59T^{2} \)
61 \( 1 + 6.66T + 61T^{2} \)
67 \( 1 - 3.73T + 67T^{2} \)
71 \( 1 + 1.80T + 71T^{2} \)
73 \( 1 + 4.46T + 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 - 0.267T + 83T^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + 9.12T + 97T^{2} \)
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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

−7.74125994588042306531232476338, −6.87176852707606962972417218936, −6.12538314503489766221221233957, −5.41497128270976921360655609535, −4.69346353739913181874572175716, −3.84457553767279559369535055902, −2.92328063462744528671562444541, −1.95314250417126341780329729659, −1.32259531930289380905985966900, 0, 1.32259531930289380905985966900, 1.95314250417126341780329729659, 2.92328063462744528671562444541, 3.84457553767279559369535055902, 4.69346353739913181874572175716, 5.41497128270976921360655609535, 6.12538314503489766221221233957, 6.87176852707606962972417218936, 7.74125994588042306531232476338

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