Properties

Label 2-7260-1.1-c1-0-6
Degree $2$
Conductor $7260$
Sign $1$
Analytic cond. $57.9713$
Root an. cond. $7.61389$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 2·7-s + 9-s − 2·13-s − 15-s − 2·19-s − 2·21-s + 25-s + 27-s + 8·31-s + 2·35-s + 2·37-s − 2·39-s − 2·43-s − 45-s − 3·49-s + 6·53-s − 2·57-s − 12·59-s − 2·61-s − 2·63-s + 2·65-s − 4·67-s − 2·73-s + 75-s + 10·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s − 0.458·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.43·31-s + 0.338·35-s + 0.328·37-s − 0.320·39-s − 0.304·43-s − 0.149·45-s − 3/7·49-s + 0.824·53-s − 0.264·57-s − 1.56·59-s − 0.256·61-s − 0.251·63-s + 0.248·65-s − 0.488·67-s − 0.234·73-s + 0.115·75-s + 1.12·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7260\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(57.9713\)
Root analytic conductor: \(7.61389\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7260,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.709283426\)
\(L(\frac12)\) \(\approx\) \(1.709283426\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.923407605983994250589950629717, −7.30006072679712949741046823488, −6.57103895258206465198836329947, −5.99832161434329141032732583940, −4.87760095706776241965692070574, −4.33147484644671132179144222503, −3.41292940185586984061180216922, −2.85372106253460268189300561756, −1.94220370965866716469868347087, −0.62346580820178742493032848749, 0.62346580820178742493032848749, 1.94220370965866716469868347087, 2.85372106253460268189300561756, 3.41292940185586984061180216922, 4.33147484644671132179144222503, 4.87760095706776241965692070574, 5.99832161434329141032732583940, 6.57103895258206465198836329947, 7.30006072679712949741046823488, 7.923407605983994250589950629717

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