Properties

Label 2-8624-1.1-c1-0-132
Degree $2$
Conductor $8624$
Sign $-1$
Analytic cond. $68.8629$
Root an. cond. $8.29837$
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  − 1.49·3-s + 1.26·5-s − 0.770·9-s + 11-s − 6.62·13-s − 1.88·15-s + 1.02·17-s − 2.10·19-s + 1.84·23-s − 3.40·25-s + 5.63·27-s + 4.86·29-s + 1.60·31-s − 1.49·33-s − 2.16·37-s + 9.88·39-s + 6.49·41-s + 12.3·43-s − 0.974·45-s + 5.02·47-s − 1.53·51-s + 2.52·53-s + 1.26·55-s + 3.13·57-s − 3.54·59-s + 2.94·61-s − 8.36·65-s + ⋯
L(s)  = 1  − 0.861·3-s + 0.565·5-s − 0.256·9-s + 0.301·11-s − 1.83·13-s − 0.487·15-s + 0.248·17-s − 0.482·19-s + 0.385·23-s − 0.680·25-s + 1.08·27-s + 0.903·29-s + 0.289·31-s − 0.259·33-s − 0.355·37-s + 1.58·39-s + 1.01·41-s + 1.88·43-s − 0.145·45-s + 0.733·47-s − 0.214·51-s + 0.346·53-s + 0.170·55-s + 0.415·57-s − 0.460·59-s + 0.377·61-s − 1.03·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8624\)    =    \(2^{4} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(68.8629\)
Root analytic conductor: \(8.29837\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8624,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + 1.49T + 3T^{2} \)
5 \( 1 - 1.26T + 5T^{2} \)
13 \( 1 + 6.62T + 13T^{2} \)
17 \( 1 - 1.02T + 17T^{2} \)
19 \( 1 + 2.10T + 19T^{2} \)
23 \( 1 - 1.84T + 23T^{2} \)
29 \( 1 - 4.86T + 29T^{2} \)
31 \( 1 - 1.60T + 31T^{2} \)
37 \( 1 + 2.16T + 37T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 - 5.02T + 47T^{2} \)
53 \( 1 - 2.52T + 53T^{2} \)
59 \( 1 + 3.54T + 59T^{2} \)
61 \( 1 - 2.94T + 61T^{2} \)
67 \( 1 - 9.84T + 67T^{2} \)
71 \( 1 + 8.86T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 - 8.31T + 79T^{2} \)
83 \( 1 + 4.54T + 83T^{2} \)
89 \( 1 + 6.62T + 89T^{2} \)
97 \( 1 + 1.10T + 97T^{2} \)
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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

−7.29070410930602769555610865643, −6.69065713629950602204171383386, −5.90839471544056526543034140835, −5.49653793262762810085980324564, −4.73591945731191907365039470785, −4.11772178022066034019138018331, −2.79781902170812333856919235267, −2.34222466721119668011254353231, −1.07893465019443229830299615489, 0, 1.07893465019443229830299615489, 2.34222466721119668011254353231, 2.79781902170812333856919235267, 4.11772178022066034019138018331, 4.73591945731191907365039470785, 5.49653793262762810085980324564, 5.90839471544056526543034140835, 6.69065713629950602204171383386, 7.29070410930602769555610865643

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