Properties

Label 2-240-4.3-c2-0-5
Degree $2$
Conductor $240$
Sign $i$
Analytic cond. $6.53952$
Root an. cond. $2.55724$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s − 2.23·5-s − 11.2i·7-s − 2.99·9-s − 4.28i·11-s − 21.4·13-s − 3.87i·15-s + 25.4·17-s − 22.4i·19-s + 19.4·21-s − 37.9i·23-s + 5.00·25-s − 5.19i·27-s + 1.41·29-s + 19.1i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.447·5-s − 1.60i·7-s − 0.333·9-s − 0.389i·11-s − 1.64·13-s − 0.258i·15-s + 1.49·17-s − 1.18i·19-s + 0.924·21-s − 1.64i·23-s + 0.200·25-s − 0.192i·27-s + 0.0488·29-s + 0.617i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $i$
Analytic conductor: \(6.53952\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.703160 - 0.703160i\)
\(L(\frac12)\) \(\approx\) \(0.703160 - 0.703160i\)
\(L(2)\) 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 \( 1 - 1.73iT \)
5 \( 1 + 2.23T \)
good7 \( 1 + 11.2iT - 49T^{2} \)
11 \( 1 + 4.28iT - 121T^{2} \)
13 \( 1 + 21.4T + 169T^{2} \)
17 \( 1 - 25.4T + 289T^{2} \)
19 \( 1 + 22.4iT - 361T^{2} \)
23 \( 1 + 37.9iT - 529T^{2} \)
29 \( 1 - 1.41T + 841T^{2} \)
31 \( 1 - 19.1iT - 961T^{2} \)
37 \( 1 + 20.2T + 1.36e3T^{2} \)
41 \( 1 + 30T + 1.68e3T^{2} \)
43 \( 1 - 24.0iT - 1.84e3T^{2} \)
47 \( 1 + 70.9iT - 2.20e3T^{2} \)
53 \( 1 - 64.2T + 2.80e3T^{2} \)
59 \( 1 - 88.6iT - 3.48e3T^{2} \)
61 \( 1 + 66.4T + 3.72e3T^{2} \)
67 \( 1 + 36.6iT - 4.48e3T^{2} \)
71 \( 1 - 133. iT - 5.04e3T^{2} \)
73 \( 1 + 28.8T + 5.32e3T^{2} \)
79 \( 1 + 60.7iT - 6.24e3T^{2} \)
83 \( 1 - 4.90iT - 6.88e3T^{2} \)
89 \( 1 - 32.8T + 7.92e3T^{2} \)
97 \( 1 + 14T + 9.40e3T^{2} \)
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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

−11.60581601888960075930464708715, −10.39757107226034085239541815090, −10.13188674611202100494526033412, −8.743062134363354185137256413998, −7.57458445716484285118464006591, −6.86598106796148264560872493952, −5.11662393693540468556568635754, −4.24343283940517143716101406408, −3.01732619337867202686063809411, −0.52142987771127098182874881750, 1.93185878518049828233458046547, 3.25127019077945428682781556632, 5.11867077292626081941876076488, 5.90058748072078028904481110090, 7.39681379910880963196435175438, 8.003355003461797680419747308944, 9.262858306786431619088551960118, 10.04214222107677714932973958292, 11.67630963319132105812066959329, 12.14796666239404904199259604030

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