Properties

Label 2-8624-1.1-c1-0-130
Degree $2$
Conductor $8624$
Sign $-1$
Analytic cond. $68.8629$
Root an. cond. $8.29837$
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 + 2·5-s + 6·9-s + 11-s − 7·13-s − 6·15-s + 2·17-s + 8·23-s − 25-s − 9·27-s − 5·29-s − 4·31-s − 3·33-s + 4·37-s + 21·39-s + 4·41-s + 8·43-s + 12·45-s − 2·47-s − 6·51-s − 6·53-s + 2·55-s − 3·59-s + 61-s − 14·65-s − 9·67-s − 24·69-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.894·5-s + 2·9-s + 0.301·11-s − 1.94·13-s − 1.54·15-s + 0.485·17-s + 1.66·23-s − 1/5·25-s − 1.73·27-s − 0.928·29-s − 0.718·31-s − 0.522·33-s + 0.657·37-s + 3.36·39-s + 0.624·41-s + 1.21·43-s + 1.78·45-s − 0.291·47-s − 0.840·51-s − 0.824·53-s + 0.269·55-s − 0.390·59-s + 0.128·61-s − 1.73·65-s − 1.09·67-s − 2.88·69-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: yes
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 \( 1 + p T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−7.24138112838676071529877740322, −6.67946617283299798598750985856, −5.83912093948941742699162815955, −5.49383997438053223488269666836, −4.85891617558259709174116216939, −4.24676377175376547806194673408, −2.96381076072196429827857946976, −2.00336255656810805561731613280, −1.07696943335018709894615475359, 0, 1.07696943335018709894615475359, 2.00336255656810805561731613280, 2.96381076072196429827857946976, 4.24676377175376547806194673408, 4.85891617558259709174116216939, 5.49383997438053223488269666836, 5.83912093948941742699162815955, 6.67946617283299798598750985856, 7.24138112838676071529877740322

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