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Properties

Label 2-240-15.14-c8-0-41
Degree $2$
Conductor $240$
Sign $-0.826 + 0.562i$
Analytic cond. $97.7708$
Root an. cond. $9.88791$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.4 + 80.0i)3-s + (268. + 564. i)5-s + 1.18e3i·7-s + (−6.25e3 + 1.98e3i)9-s + 2.05e4i·11-s + 5.29e4i·13-s + (−4.18e4 + 2.84e4i)15-s + 2.52e4·17-s − 1.11e5·19-s + (−9.45e4 + 1.46e4i)21-s + 3.46e5·23-s + (−2.46e5 + 3.02e5i)25-s + (−2.36e5 − 4.75e5i)27-s + 8.59e4i·29-s + 5.62e5·31-s + ⋯
L(s)  = 1  + (0.153 + 0.988i)3-s + (0.429 + 0.903i)5-s + 0.491i·7-s + (−0.952 + 0.303i)9-s + 1.40i·11-s + 1.85i·13-s + (−0.826 + 0.562i)15-s + 0.302·17-s − 0.858·19-s + (−0.485 + 0.0754i)21-s + 1.23·23-s + (−0.631 + 0.775i)25-s + (−0.445 − 0.895i)27-s + 0.121i·29-s + 0.609·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.826 + 0.562i$
Analytic conductor: \(97.7708\)
Root analytic conductor: \(9.88791\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :4),\ -0.826 + 0.562i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.287772984\)
\(L(\frac12)\) \(\approx\) \(2.287772984\)
\(L(5)\) 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 \( 1 + (-12.4 - 80.0i)T \)
5 \( 1 + (-268. - 564. i)T \)
good7 \( 1 - 1.18e3iT - 5.76e6T^{2} \)
11 \( 1 - 2.05e4iT - 2.14e8T^{2} \)
13 \( 1 - 5.29e4iT - 8.15e8T^{2} \)
17 \( 1 - 2.52e4T + 6.97e9T^{2} \)
19 \( 1 + 1.11e5T + 1.69e10T^{2} \)
23 \( 1 - 3.46e5T + 7.83e10T^{2} \)
29 \( 1 - 8.59e4iT - 5.00e11T^{2} \)
31 \( 1 - 5.62e5T + 8.52e11T^{2} \)
37 \( 1 - 5.55e5iT - 3.51e12T^{2} \)
41 \( 1 - 4.89e6iT - 7.98e12T^{2} \)
43 \( 1 - 2.65e5iT - 1.16e13T^{2} \)
47 \( 1 + 2.91e6T + 2.38e13T^{2} \)
53 \( 1 - 1.22e7T + 6.22e13T^{2} \)
59 \( 1 - 5.62e6iT - 1.46e14T^{2} \)
61 \( 1 - 2.68e7T + 1.91e14T^{2} \)
67 \( 1 + 2.98e7iT - 4.06e14T^{2} \)
71 \( 1 + 8.69e6iT - 6.45e14T^{2} \)
73 \( 1 - 2.60e7iT - 8.06e14T^{2} \)
79 \( 1 + 9.67e6T + 1.51e15T^{2} \)
83 \( 1 + 1.32e7T + 2.25e15T^{2} \)
89 \( 1 + 9.04e7iT - 3.93e15T^{2} \)
97 \( 1 - 8.87e7iT - 7.83e15T^{2} \)
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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

−11.18546387157685846450860901029, −10.11762931810215005335761420159, −9.526491223313105822843409590709, −8.648677840055744166259693737652, −7.14504085237027256692329707390, −6.31969934964765894089702581259, −4.97112898655498353519993782566, −4.10086882492040437563164660764, −2.74881062148503296784237541681, −1.87295914716061366594614667088, 0.62065653625975027508528592643, 0.859605219287211733172823912800, 2.44426790156778330997117738946, 3.58317320463919108555301407476, 5.31775024117739731298529735289, 5.93732307375092526257661378982, 7.21240492416906349107201679929, 8.339692002775295816554923978060, 8.710474599905584287624492953178, 10.19882729718336658854624499620

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