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Properties

Label 2-240-15.14-c2-0-18
Degree $2$
Conductor $240$
Sign $0.290 + 0.956i$
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  + (2.72 − 1.26i)3-s + (0.689 − 4.95i)5-s − 0.735i·7-s + (5.82 − 6.86i)9-s + 10.9i·11-s − 21.1i·13-s + (−4.36 − 14.3i)15-s + 7.03·17-s − 23.1·19-s + (−0.927 − 2.00i)21-s + 24.7·23-s + (−24.0 − 6.82i)25-s + (7.21 − 26.0i)27-s + 32.3i·29-s + 34.9·31-s + ⋯
L(s)  = 1  + (0.907 − 0.420i)3-s + (0.137 − 0.990i)5-s − 0.105i·7-s + (0.647 − 0.762i)9-s + 0.995i·11-s − 1.63i·13-s + (−0.290 − 0.956i)15-s + 0.413·17-s − 1.21·19-s + (−0.0441 − 0.0953i)21-s + 1.07·23-s + (−0.962 − 0.272i)25-s + (0.267 − 0.963i)27-s + 1.11i·29-s + 1.12·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.290 + 0.956i$
Analytic conductor: \(6.53952\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1),\ 0.290 + 0.956i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.70071 - 1.26042i\)
\(L(\frac12)\) \(\approx\) \(1.70071 - 1.26042i\)
\(L(2)\) 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 \( 1 + (-2.72 + 1.26i)T \)
5 \( 1 + (-0.689 + 4.95i)T \)
good7 \( 1 + 0.735iT - 49T^{2} \)
11 \( 1 - 10.9iT - 121T^{2} \)
13 \( 1 + 21.1iT - 169T^{2} \)
17 \( 1 - 7.03T + 289T^{2} \)
19 \( 1 + 23.1T + 361T^{2} \)
23 \( 1 - 24.7T + 529T^{2} \)
29 \( 1 - 32.3iT - 841T^{2} \)
31 \( 1 - 34.9T + 961T^{2} \)
37 \( 1 + 37.7iT - 1.36e3T^{2} \)
41 \( 1 - 39.0iT - 1.68e3T^{2} \)
43 \( 1 - 22.6iT - 1.84e3T^{2} \)
47 \( 1 + 39.1T + 2.20e3T^{2} \)
53 \( 1 - 60.9T + 2.80e3T^{2} \)
59 \( 1 - 7.79iT - 3.48e3T^{2} \)
61 \( 1 + 11.1T + 3.72e3T^{2} \)
67 \( 1 - 33.3iT - 4.48e3T^{2} \)
71 \( 1 - 96.9iT - 5.04e3T^{2} \)
73 \( 1 - 134. iT - 5.32e3T^{2} \)
79 \( 1 + 121.T + 6.24e3T^{2} \)
83 \( 1 - 90.2T + 6.88e3T^{2} \)
89 \( 1 - 53.1iT - 7.92e3T^{2} \)
97 \( 1 - 115. iT - 9.40e3T^{2} \)
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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

−12.11889361863236319537951581633, −10.51327971036299396142305703012, −9.653800432760662377675457586002, −8.640671075220997165557870266970, −7.947978294316490142004614910041, −6.87688662876727243795424864634, −5.40274025295582675172243001471, −4.19912853520508416482519914433, −2.69305804150334810322208699481, −1.13153909278530656020438027361, 2.16360506762907009292377625240, 3.33616271212771172498748246617, 4.47300513952570861959167376055, 6.16812067144814244239224753255, 7.11534549136330142561051217976, 8.337979875746491290769671359596, 9.146141449365194004868150208151, 10.17101948471627224096882703597, 10.98944256544482091609638764185, 11.92820406879090015613380192215

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