Properties

Label 2-240-1.1-c9-0-32
Degree $2$
Conductor $240$
Sign $-1$
Analytic cond. $123.608$
Root an. cond. $11.1179$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 81·3-s − 625·5-s + 5.98e3·7-s + 6.56e3·9-s + 1.46e4·11-s + 3.79e4·13-s − 5.06e4·15-s − 4.41e5·17-s − 4.41e5·19-s + 4.85e5·21-s − 2.26e6·23-s + 3.90e5·25-s + 5.31e5·27-s − 1.04e6·29-s + 7.91e6·31-s + 1.18e6·33-s − 3.74e6·35-s − 2.09e7·37-s + 3.07e6·39-s + 1.32e7·41-s + 2.31e7·43-s − 4.10e6·45-s + 1.38e7·47-s − 4.49e6·49-s − 3.57e7·51-s − 5.76e7·53-s − 9.15e6·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.942·7-s + 1/3·9-s + 0.301·11-s + 0.368·13-s − 0.258·15-s − 1.28·17-s − 0.777·19-s + 0.544·21-s − 1.68·23-s + 1/5·25-s + 0.192·27-s − 0.275·29-s + 1.53·31-s + 0.174·33-s − 0.421·35-s − 1.84·37-s + 0.212·39-s + 0.734·41-s + 1.03·43-s − 0.149·45-s + 0.414·47-s − 0.111·49-s − 0.739·51-s − 1.00·53-s − 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-1$
Analytic conductor: \(123.608\)
Root analytic conductor: \(11.1179\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 240,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - p^{4} T \)
5 \( 1 + p^{4} T \)
good7 \( 1 - 5988 T + p^{9} T^{2} \)
11 \( 1 - 14648 T + p^{9} T^{2} \)
13 \( 1 - 37906 T + p^{9} T^{2} \)
17 \( 1 + 441098 T + p^{9} T^{2} \)
19 \( 1 + 441820 T + p^{9} T^{2} \)
23 \( 1 + 2264136 T + p^{9} T^{2} \)
29 \( 1 + 1049350 T + p^{9} T^{2} \)
31 \( 1 - 7910568 T + p^{9} T^{2} \)
37 \( 1 + 20992558 T + p^{9} T^{2} \)
41 \( 1 - 13285562 T + p^{9} T^{2} \)
43 \( 1 - 23130764 T + p^{9} T^{2} \)
47 \( 1 - 13873688 T + p^{9} T^{2} \)
53 \( 1 + 57635174 T + p^{9} T^{2} \)
59 \( 1 - 32042120 T + p^{9} T^{2} \)
61 \( 1 - 110664022 T + p^{9} T^{2} \)
67 \( 1 - 118568268 T + p^{9} T^{2} \)
71 \( 1 + 276679712 T + p^{9} T^{2} \)
73 \( 1 + 264023294 T + p^{9} T^{2} \)
79 \( 1 + 448202760 T + p^{9} T^{2} \)
83 \( 1 + 851015796 T + p^{9} T^{2} \)
89 \( 1 - 189894930 T + p^{9} T^{2} \)
97 \( 1 + 1014149278 T + p^{9} T^{2} \)
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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

−10.05872383903168484858507202147, −8.733280668910766944971730405459, −8.289930487500002701080423217688, −7.21077797614437005522624556743, −6.08693858078281104687080140454, −4.58447557963956180798183097238, −3.93042063588217633109434125226, −2.45065785132845484347453635035, −1.48498491562208623158383891692, 0, 1.48498491562208623158383891692, 2.45065785132845484347453635035, 3.93042063588217633109434125226, 4.58447557963956180798183097238, 6.08693858078281104687080140454, 7.21077797614437005522624556743, 8.289930487500002701080423217688, 8.733280668910766944971730405459, 10.05872383903168484858507202147

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