Properties

Label 2-240-1.1-c3-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 5·5-s + 24·7-s + 9·9-s − 52·11-s + 22·13-s − 15·15-s − 14·17-s + 20·19-s − 72·21-s + 168·23-s + 25·25-s − 27·27-s + 230·29-s + 288·31-s + 156·33-s + 120·35-s − 34·37-s − 66·39-s + 122·41-s + 188·43-s + 45·45-s − 256·47-s + 233·49-s + 42·51-s − 338·53-s − 260·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.29·7-s + 1/3·9-s − 1.42·11-s + 0.469·13-s − 0.258·15-s − 0.199·17-s + 0.241·19-s − 0.748·21-s + 1.52·23-s + 1/5·25-s − 0.192·27-s + 1.47·29-s + 1.66·31-s + 0.822·33-s + 0.579·35-s − 0.151·37-s − 0.270·39-s + 0.464·41-s + 0.666·43-s + 0.149·45-s − 0.794·47-s + 0.679·49-s + 0.115·51-s − 0.875·53-s − 0.637·55-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: $\chi_{240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.794111648\)
\(L(\frac12)\) \(\approx\) \(1.794111648\)
\(L(\frac{5}{2})\) 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 \( 1 - 24 T + p^{3} T^{2} \)
11 \( 1 + 52 T + p^{3} T^{2} \)
13 \( 1 - 22 T + p^{3} T^{2} \)
17 \( 1 + 14 T + p^{3} T^{2} \)
19 \( 1 - 20 T + p^{3} T^{2} \)
23 \( 1 - 168 T + p^{3} T^{2} \)
29 \( 1 - 230 T + p^{3} T^{2} \)
31 \( 1 - 288 T + p^{3} T^{2} \)
37 \( 1 + 34 T + p^{3} T^{2} \)
41 \( 1 - 122 T + p^{3} T^{2} \)
43 \( 1 - 188 T + p^{3} T^{2} \)
47 \( 1 + 256 T + p^{3} T^{2} \)
53 \( 1 + 338 T + p^{3} T^{2} \)
59 \( 1 + 100 T + p^{3} T^{2} \)
61 \( 1 - 742 T + p^{3} T^{2} \)
67 \( 1 - 84 T + p^{3} T^{2} \)
71 \( 1 - 328 T + p^{3} T^{2} \)
73 \( 1 + 38 T + p^{3} T^{2} \)
79 \( 1 - 240 T + p^{3} T^{2} \)
83 \( 1 + 1212 T + p^{3} T^{2} \)
89 \( 1 - 330 T + p^{3} T^{2} \)
97 \( 1 - 866 T + p^{3} T^{2} \)
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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

−11.47801404649029448443985535584, −10.83034202748639418595821720198, −10.01071311566315588266155097002, −8.606965907592145758127059846040, −7.79399895337510396010165449570, −6.54732761886142696865518917324, −5.29367848302165795090701896795, −4.67946886581896779290168536334, −2.66696360287767869903250095831, −1.08770567105234639276599270112, 1.08770567105234639276599270112, 2.66696360287767869903250095831, 4.67946886581896779290168536334, 5.29367848302165795090701896795, 6.54732761886142696865518917324, 7.79399895337510396010165449570, 8.606965907592145758127059846040, 10.01071311566315588266155097002, 10.83034202748639418595821720198, 11.47801404649029448443985535584

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