Properties

Label 2-9016-1.1-c1-0-139
Degree $2$
Conductor $9016$
Sign $-1$
Analytic cond. $71.9931$
Root an. cond. $8.48487$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 6·9-s + 5·13-s + 6·17-s − 6·19-s + 23-s − 5·25-s − 9·27-s + 9·29-s − 3·31-s − 8·37-s − 15·39-s − 3·41-s − 8·43-s − 7·47-s − 18·51-s − 2·53-s + 18·57-s − 4·59-s + 10·61-s + 8·67-s − 3·69-s + 7·71-s − 9·73-s + 15·75-s − 6·79-s + 9·81-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s + 1.38·13-s + 1.45·17-s − 1.37·19-s + 0.208·23-s − 25-s − 1.73·27-s + 1.67·29-s − 0.538·31-s − 1.31·37-s − 2.40·39-s − 0.468·41-s − 1.21·43-s − 1.02·47-s − 2.52·51-s − 0.274·53-s + 2.38·57-s − 0.520·59-s + 1.28·61-s + 0.977·67-s − 0.361·69-s + 0.830·71-s − 1.05·73-s + 1.73·75-s − 0.675·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9016\)    =    \(2^{3} \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(71.9931\)
Root analytic conductor: \(8.48487\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9016,\ (\ :1/2),\ -1)\)

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 \( 1 \)
23 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−7.04197497337152687820054834122, −6.58816593954710107070945478976, −5.99955004194896901651448781012, −5.43923956112430414616921048860, −4.80224422790616237388517868450, −3.97526882511140771304727543873, −3.29167515147874786207239858147, −1.80587464224290004012260590916, −1.08088799825528163961182664494, 0, 1.08088799825528163961182664494, 1.80587464224290004012260590916, 3.29167515147874786207239858147, 3.97526882511140771304727543873, 4.80224422790616237388517868450, 5.43923956112430414616921048860, 5.99955004194896901651448781012, 6.58816593954710107070945478976, 7.04197497337152687820054834122

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