Properties

Label 2-12480-1.1-c1-0-62
Degree $2$
Conductor $12480$
Sign $-1$
Analytic cond. $99.6533$
Root an. cond. $9.98265$
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-s + 5-s + 9-s + 13-s − 15-s + 2·17-s + 4·23-s + 25-s − 27-s − 6·29-s + 8·31-s − 10·37-s − 39-s − 6·41-s − 4·43-s + 45-s − 8·47-s − 7·49-s − 2·51-s − 2·53-s − 6·61-s + 65-s − 12·67-s − 4·69-s + 8·71-s + 6·73-s − 75-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 0.485·17-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 1.64·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s − 49-s − 0.280·51-s − 0.274·53-s − 0.768·61-s + 0.124·65-s − 1.46·67-s − 0.481·69-s + 0.949·71-s + 0.702·73-s − 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12480\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(99.6533\)
Root analytic conductor: \(9.98265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 12480,\ (\ :1/2),\ -1)\)

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

−16.72703744639065, −16.16652884049546, −15.35518303965535, −15.10144190152596, −14.29300393641957, −13.65551701100799, −13.28152938324184, −12.56212330624923, −12.05658556026194, −11.44546600120969, −10.85654851515303, −10.30435331447386, −9.715501622567799, −9.162531466435019, −8.371654595122781, −7.833356249553535, −6.888398243766097, −6.569955102215190, −5.785067755082558, −5.138947790489691, −4.677151344697214, −3.607010159437114, −3.045208817578382, −1.899537738679919, −1.226547260975463, 0, 1.226547260975463, 1.899537738679919, 3.045208817578382, 3.607010159437114, 4.677151344697214, 5.138947790489691, 5.785067755082558, 6.569955102215190, 6.888398243766097, 7.833356249553535, 8.371654595122781, 9.162531466435019, 9.715501622567799, 10.30435331447386, 10.85654851515303, 11.44546600120969, 12.05658556026194, 12.56212330624923, 13.28152938324184, 13.65551701100799, 14.29300393641957, 15.10144190152596, 15.35518303965535, 16.16652884049546, 16.72703744639065

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