Properties

Label 2-240-15.14-c2-0-6
Degree $2$
Conductor $240$
Sign $1$
Analytic cond. $6.53952$
Root an. cond. $2.55724$
Motivic weight $2$
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 + 9·9-s + 15·15-s + 14·17-s + 22·19-s + 34·23-s + 25·25-s − 27·27-s − 2·31-s − 45·45-s − 14·47-s + 49·49-s − 42·51-s + 86·53-s − 66·57-s − 118·61-s − 102·69-s − 75·75-s − 98·79-s + 81·81-s + 154·83-s − 70·85-s + 6·93-s − 110·95-s + 106·107-s − 22·109-s + ⋯
L(s)  = 1  − 3-s − 5-s + 9-s + 15-s + 0.823·17-s + 1.15·19-s + 1.47·23-s + 25-s − 27-s − 0.0645·31-s − 45-s − 0.297·47-s + 49-s − 0.823·51-s + 1.62·53-s − 1.15·57-s − 1.93·61-s − 1.47·69-s − 75-s − 1.24·79-s + 81-s + 1.85·83-s − 0.823·85-s + 2/31·93-s − 1.15·95-s + 0.990·107-s − 0.201·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(6.53952\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{240} (209, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9500007074\)
\(L(\frac12)\) \(\approx\) \(0.9500007074\)
\(L(2)\) 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 \( ( 1 - p T )( 1 + p T ) \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( ( 1 - p T )( 1 + p T ) \)
17 \( 1 - 14 T + p^{2} T^{2} \)
19 \( 1 - 22 T + p^{2} T^{2} \)
23 \( 1 - 34 T + p^{2} T^{2} \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 + 2 T + p^{2} T^{2} \)
37 \( ( 1 - p T )( 1 + p T ) \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( ( 1 - p T )( 1 + p T ) \)
47 \( 1 + 14 T + p^{2} T^{2} \)
53 \( 1 - 86 T + p^{2} T^{2} \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 + 118 T + p^{2} T^{2} \)
67 \( ( 1 - p T )( 1 + p T ) \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( ( 1 - p T )( 1 + p T ) \)
79 \( 1 + 98 T + p^{2} T^{2} \)
83 \( 1 - 154 T + p^{2} T^{2} \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( ( 1 - p T )( 1 + p T ) \)
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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

−11.86156989113948604228214240712, −11.11445713999578317349934517246, −10.21690467396352404198008924225, −9.033959945069231142750103527307, −7.66967602596282703117398136086, −6.99996739938410602676470067953, −5.63106332156228966461888256364, −4.65103420922866794220530252139, −3.34658018088129583030645705346, −0.910310005925037530481401702831, 0.910310005925037530481401702831, 3.34658018088129583030645705346, 4.65103420922866794220530252139, 5.63106332156228966461888256364, 6.99996739938410602676470067953, 7.66967602596282703117398136086, 9.033959945069231142750103527307, 10.21690467396352404198008924225, 11.11445713999578317349934517246, 11.86156989113948604228214240712

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