Properties

Label 2-240-1.1-c3-0-8
Degree $2$
Conductor $240$
Sign $-1$
Analytic cond. $14.1604$
Root an. cond. $3.76303$
Motivic weight $3$
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 + 5·5-s − 8·7-s + 9·9-s − 20·11-s + 22·13-s − 15·15-s − 14·17-s − 76·19-s + 24·21-s − 56·23-s + 25·25-s − 27·27-s − 154·29-s − 160·31-s + 60·33-s − 40·35-s − 162·37-s − 66·39-s − 390·41-s − 388·43-s + 45·45-s + 544·47-s − 279·49-s + 42·51-s − 210·53-s − 100·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.431·7-s + 1/3·9-s − 0.548·11-s + 0.469·13-s − 0.258·15-s − 0.199·17-s − 0.917·19-s + 0.249·21-s − 0.507·23-s + 1/5·25-s − 0.192·27-s − 0.986·29-s − 0.926·31-s + 0.316·33-s − 0.193·35-s − 0.719·37-s − 0.270·39-s − 1.48·41-s − 1.37·43-s + 0.149·45-s + 1.68·47-s − 0.813·49-s + 0.115·51-s − 0.544·53-s − 0.245·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-1$
Analytic conductor: \(14.1604\)
Root analytic conductor: \(3.76303\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 240,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
5 \( 1 - p T \)
good7 \( 1 + 8 T + p^{3} T^{2} \)
11 \( 1 + 20 T + p^{3} T^{2} \)
13 \( 1 - 22 T + p^{3} T^{2} \)
17 \( 1 + 14 T + p^{3} T^{2} \)
19 \( 1 + 4 p T + p^{3} T^{2} \)
23 \( 1 + 56 T + p^{3} T^{2} \)
29 \( 1 + 154 T + p^{3} T^{2} \)
31 \( 1 + 160 T + p^{3} T^{2} \)
37 \( 1 + 162 T + p^{3} T^{2} \)
41 \( 1 + 390 T + p^{3} T^{2} \)
43 \( 1 + 388 T + p^{3} T^{2} \)
47 \( 1 - 544 T + p^{3} T^{2} \)
53 \( 1 + 210 T + p^{3} T^{2} \)
59 \( 1 - 380 T + p^{3} T^{2} \)
61 \( 1 + 794 T + p^{3} T^{2} \)
67 \( 1 - 148 T + p^{3} T^{2} \)
71 \( 1 - 840 T + p^{3} T^{2} \)
73 \( 1 - 858 T + p^{3} T^{2} \)
79 \( 1 + 144 T + p^{3} T^{2} \)
83 \( 1 + 316 T + p^{3} T^{2} \)
89 \( 1 - 1098 T + p^{3} T^{2} \)
97 \( 1 - 994 T + p^{3} T^{2} \)
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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

−11.06717424126727598242449902628, −10.37155012756565220703745257976, −9.385709448737840233568650887911, −8.279653816861754633348086696081, −6.96944553952151673201759704377, −6.05821022984430241169824054005, −5.06784739914740818575535604282, −3.62532930215508665042642724423, −1.92985538548215484025166663313, 0, 1.92985538548215484025166663313, 3.62532930215508665042642724423, 5.06784739914740818575535604282, 6.05821022984430241169824054005, 6.96944553952151673201759704377, 8.279653816861754633348086696081, 9.385709448737840233568650887911, 10.37155012756565220703745257976, 11.06717424126727598242449902628

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