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Properties

Label 2-240-15.14-c2-0-17
Degree $2$
Conductor $240$
Sign $0.395 + 0.918i$
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  + (2.49 + 1.67i)3-s + (−4.19 − 2.71i)5-s − 12.7i·7-s + (3.41 + 8.32i)9-s − 12.6i·11-s − 7.44i·13-s + (−5.92 − 13.7i)15-s − 14.0·17-s + 31.0·19-s + (21.3 − 31.8i)21-s − 7.50·23-s + (10.2 + 22.7i)25-s + (−5.40 + 26.4i)27-s + 15.7i·29-s + 20.4·31-s + ⋯
L(s)  = 1  + (0.830 + 0.557i)3-s + (−0.839 − 0.542i)5-s − 1.82i·7-s + (0.379 + 0.925i)9-s − 1.14i·11-s − 0.572i·13-s + (−0.395 − 0.918i)15-s − 0.826·17-s + 1.63·19-s + (1.01 − 1.51i)21-s − 0.326·23-s + (0.410 + 0.911i)25-s + (−0.200 + 0.979i)27-s + 0.542i·29-s + 0.660·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.395 + 0.918i$
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: \((2,\ 240,\ (\ :1),\ 0.395 + 0.918i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.37359 - 0.904526i\)
\(L(\frac12)\) \(\approx\) \(1.37359 - 0.904526i\)
\(L(2)\) 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 \( 1 + (-2.49 - 1.67i)T \)
5 \( 1 + (4.19 + 2.71i)T \)
good7 \( 1 + 12.7iT - 49T^{2} \)
11 \( 1 + 12.6iT - 121T^{2} \)
13 \( 1 + 7.44iT - 169T^{2} \)
17 \( 1 + 14.0T + 289T^{2} \)
19 \( 1 - 31.0T + 361T^{2} \)
23 \( 1 + 7.50T + 529T^{2} \)
29 \( 1 - 15.7iT - 841T^{2} \)
31 \( 1 - 20.4T + 961T^{2} \)
37 \( 1 + 12.9iT - 1.36e3T^{2} \)
41 \( 1 + 13.8iT - 1.68e3T^{2} \)
43 \( 1 + 30.0iT - 1.84e3T^{2} \)
47 \( 1 + 20.2T + 2.20e3T^{2} \)
53 \( 1 - 29.1T + 2.80e3T^{2} \)
59 \( 1 + 47.6iT - 3.48e3T^{2} \)
61 \( 1 - 43.0T + 3.72e3T^{2} \)
67 \( 1 + 0.630iT - 4.48e3T^{2} \)
71 \( 1 - 90.4iT - 5.04e3T^{2} \)
73 \( 1 - 46.2iT - 5.32e3T^{2} \)
79 \( 1 + 37.9T + 6.24e3T^{2} \)
83 \( 1 - 80.2T + 6.88e3T^{2} \)
89 \( 1 - 140. iT - 7.92e3T^{2} \)
97 \( 1 + 10.3iT - 9.40e3T^{2} \)
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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

−11.46914292924839971369525585447, −10.73775394351064308203730868494, −9.830432690392356866006048649629, −8.662292443598240013004025613681, −7.87115581036101533564004715681, −7.08491323691336804593194901539, −5.14697760175247816388442810760, −4.01465032341126409334536682949, −3.30990185709271309187932579374, −0.832475359297004753541465432870, 2.08959042348337828766990075509, 3.08534475088008722864632223255, 4.60760288304117403375783462209, 6.21818462105892738062630338489, 7.22146260922431598767273182225, 8.139573860455584654321284496712, 9.071405659234160782545052440830, 9.855271915726426379532908427602, 11.68840980455733122012805640457, 11.89248845808431809517180470328

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