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Properties

Label 2-240-15.14-c2-0-21
Degree $2$
Conductor $240$
Sign $-0.934 + 0.355i$
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  + (0.707 − 2.91i)3-s + (−2.82 − 4.12i)5-s + 5.83i·7-s + (−8 − 4.12i)9-s − 16.4i·11-s + (−14.0 + 5.33i)15-s − 11.3·17-s − 12·19-s + (17 + 4.12i)21-s − 24.0·23-s + (−8.99 + 23.3i)25-s + (−17.6 + 20.4i)27-s + 32·31-s + (−48.0 − 11.6i)33-s + (24.0 − 16.4i)35-s + ⋯
L(s)  = 1  + (0.235 − 0.971i)3-s + (−0.565 − 0.824i)5-s + 0.832i·7-s + (−0.888 − 0.458i)9-s − 1.49i·11-s + (−0.934 + 0.355i)15-s − 0.665·17-s − 0.631·19-s + (0.809 + 0.196i)21-s − 1.04·23-s + (−0.359 + 0.932i)25-s + (−0.654 + 0.755i)27-s + 1.03·31-s + (−1.45 − 0.353i)33-s + (0.686 − 0.471i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.934 + 0.355i$
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: \((2,\ 240,\ (\ :1),\ -0.934 + 0.355i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.178377 - 0.971095i\)
\(L(\frac12)\) \(\approx\) \(0.178377 - 0.971095i\)
\(L(2)\) 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 \( 1 + (-0.707 + 2.91i)T \)
5 \( 1 + (2.82 + 4.12i)T \)
good7 \( 1 - 5.83iT - 49T^{2} \)
11 \( 1 + 16.4iT - 121T^{2} \)
13 \( 1 - 169T^{2} \)
17 \( 1 + 11.3T + 289T^{2} \)
19 \( 1 + 12T + 361T^{2} \)
23 \( 1 + 24.0T + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 32T + 961T^{2} \)
37 \( 1 + 23.3iT - 1.36e3T^{2} \)
41 \( 1 + 57.7iT - 1.68e3T^{2} \)
43 \( 1 + 40.8iT - 1.84e3T^{2} \)
47 \( 1 - 35.3T + 2.20e3T^{2} \)
53 \( 1 - 67.8T + 2.80e3T^{2} \)
59 \( 1 + 16.4iT - 3.48e3T^{2} \)
61 \( 1 + 16T + 3.72e3T^{2} \)
67 \( 1 + 5.83iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 116. iT - 5.32e3T^{2} \)
79 \( 1 - 72T + 6.24e3T^{2} \)
83 \( 1 - 43.8T + 6.88e3T^{2} \)
89 \( 1 + 65.9iT - 7.92e3T^{2} \)
97 \( 1 - 163. iT - 9.40e3T^{2} \)
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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

−11.85800381133044555897973961326, −10.77554310695504790355079426668, −8.947657051201536855546528010379, −8.638606284034232253706828129191, −7.69838798651978055069919606691, −6.31568962816539996506819106434, −5.46442554630693547659182853113, −3.79214646943293589926134454599, −2.27096600072480738838875920779, −0.49165751528149475861300559833, 2.51930990812172090166982949046, 3.99806944319799776051741960806, 4.58988458551157477925149160282, 6.38157870445679999196167852734, 7.42071027170928327873516197194, 8.372593093865129606442053012032, 9.796047659968031750361243571807, 10.28076721636738456696965940091, 11.17733981436670646176940949400, 12.10597697591761278504808641380

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