Properties

Label 2-7280-1.1-c1-0-109
Degree $2$
Conductor $7280$
Sign $-1$
Analytic cond. $58.1310$
Root an. cond. $7.62437$
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  − 5-s + 7-s − 3·9-s − 13-s − 2·17-s + 4·19-s + 25-s − 2·29-s − 35-s + 2·37-s + 6·41-s + 4·43-s + 3·45-s + 8·47-s + 49-s + 6·53-s + 4·59-s − 10·61-s − 3·63-s + 65-s − 12·67-s − 4·71-s − 10·73-s + 9·81-s − 12·83-s + 2·85-s − 18·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 9-s − 0.277·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s − 0.169·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 0.447·45-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s − 1.28·61-s − 0.377·63-s + 0.124·65-s − 1.46·67-s − 0.474·71-s − 1.17·73-s + 81-s − 1.31·83-s + 0.216·85-s − 1.90·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7280\)    =    \(2^{4} \cdot 5 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(58.1310\)
Root analytic conductor: \(7.62437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7280,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 + T \)
good3 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.51130579904357732840659733736, −7.09919938407154638179485266362, −5.95757956225412542169589403935, −5.59587330588391857865009492155, −4.65658660100193562046579012240, −4.02721462820321817272857975691, −3.03539942710213026543664991451, −2.43327688196743013621038231227, −1.19570141677931663743820748832, 0, 1.19570141677931663743820748832, 2.43327688196743013621038231227, 3.03539942710213026543664991451, 4.02721462820321817272857975691, 4.65658660100193562046579012240, 5.59587330588391857865009492155, 5.95757956225412542169589403935, 7.09919938407154638179485266362, 7.51130579904357732840659733736

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