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Properties

Label 2-240-5.2-c2-0-8
Degree $2$
Conductor $240$
Sign $0.326 + 0.945i$
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  + (1.22 − 1.22i)3-s + (−2.67 − 4.22i)5-s + (5.44 + 5.44i)7-s − 2.99i·9-s + 6.44·11-s + (14.4 − 14.4i)13-s + (−8.44 − 1.89i)15-s + (−23.1 − 23.1i)17-s − 16.6i·19-s + 13.3·21-s + (6.65 − 6.65i)23-s + (−10.6 + 22.5i)25-s + (−3.67 − 3.67i)27-s − 0.0454i·29-s − 4.49·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.534 − 0.844i)5-s + (0.778 + 0.778i)7-s − 0.333i·9-s + 0.586·11-s + (1.11 − 1.11i)13-s + (−0.563 − 0.126i)15-s + (−1.36 − 1.36i)17-s − 0.878i·19-s + 0.635·21-s + (0.289 − 0.289i)23-s + (−0.427 + 0.903i)25-s + (−0.136 − 0.136i)27-s − 0.00156i·29-s − 0.144·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.326 + 0.945i$
Analytic conductor: \(6.53952\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1),\ 0.326 + 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.43153 - 1.02014i\)
\(L(\frac12)\) \(\approx\) \(1.43153 - 1.02014i\)
\(L(2)\) 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 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (2.67 + 4.22i)T \)
good7 \( 1 + (-5.44 - 5.44i)T + 49iT^{2} \)
11 \( 1 - 6.44T + 121T^{2} \)
13 \( 1 + (-14.4 + 14.4i)T - 169iT^{2} \)
17 \( 1 + (23.1 + 23.1i)T + 289iT^{2} \)
19 \( 1 + 16.6iT - 361T^{2} \)
23 \( 1 + (-6.65 + 6.65i)T - 529iT^{2} \)
29 \( 1 + 0.0454iT - 841T^{2} \)
31 \( 1 + 4.49T + 961T^{2} \)
37 \( 1 + (-35.3 - 35.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 20.2T + 1.68e3T^{2} \)
43 \( 1 + (32.2 - 32.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-50.5 - 50.5i)T + 2.20e3iT^{2} \)
53 \( 1 + (5.50 - 5.50i)T - 2.80e3iT^{2} \)
59 \( 1 + 55.4iT - 3.48e3T^{2} \)
61 \( 1 - 47.8T + 3.72e3T^{2} \)
67 \( 1 + (-85.2 - 85.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 48.4T + 5.04e3T^{2} \)
73 \( 1 + (21.9 - 21.9i)T - 5.32e3iT^{2} \)
79 \( 1 - 126. iT - 6.24e3T^{2} \)
83 \( 1 + (94.9 - 94.9i)T - 6.88e3iT^{2} \)
89 \( 1 + 71.7iT - 7.92e3T^{2} \)
97 \( 1 + (37 + 37i)T + 9.40e3iT^{2} \)
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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

−11.57697158517016528871969319426, −11.21113055683170589813665209595, −9.386661736721945815689127362549, −8.670293106465568213266854194545, −8.032340394165522922502340064402, −6.76625511623976274296460221651, −5.39307478154372927994305977635, −4.33494299804694941586227528324, −2.71116330706660890774172088221, −0.999571125288229702878015780125, 1.84295628538211653546036257987, 3.82162957962838370596020017406, 4.20080195956659193638343451522, 6.16265986106231581009069824232, 7.14953825064683100416716093811, 8.221133923039658916467653023909, 9.051218169451729166501859433833, 10.47153636503288189335633168565, 10.99429739172426131111505412625, 11.77316802026627499147570503440

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