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Properties

Label 2-240-15.14-c2-0-8
Degree $2$
Conductor $240$
Sign $-0.105 - 0.994i$
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  + (0.938 + 2.84i)3-s + (4.88 + 1.05i)5-s + 6.81i·7-s + (−7.23 + 5.34i)9-s + 7.52i·11-s − 16.2i·13-s + (1.58 + 14.9i)15-s + 4.11·17-s − 7.86·19-s + (−19.4 + 6.39i)21-s + 19.5·23-s + (22.7 + 10.2i)25-s + (−22.0 − 15.6i)27-s + 55.8i·29-s − 43.4·31-s + ⋯
L(s)  = 1  + (0.312 + 0.949i)3-s + (0.977 + 0.210i)5-s + 0.973i·7-s + (−0.804 + 0.594i)9-s + 0.684i·11-s − 1.24i·13-s + (0.105 + 0.994i)15-s + 0.242·17-s − 0.413·19-s + (−0.924 + 0.304i)21-s + 0.848·23-s + (0.911 + 0.411i)25-s + (−0.815 − 0.578i)27-s + 1.92i·29-s − 1.40·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.105 - 0.994i$
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: \((2,\ 240,\ (\ :1),\ -0.105 - 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.27287 + 1.41559i\)
\(L(\frac12)\) \(\approx\) \(1.27287 + 1.41559i\)
\(L(2)\) 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 \( 1 + (-0.938 - 2.84i)T \)
5 \( 1 + (-4.88 - 1.05i)T \)
good7 \( 1 - 6.81iT - 49T^{2} \)
11 \( 1 - 7.52iT - 121T^{2} \)
13 \( 1 + 16.2iT - 169T^{2} \)
17 \( 1 - 4.11T + 289T^{2} \)
19 \( 1 + 7.86T + 361T^{2} \)
23 \( 1 - 19.5T + 529T^{2} \)
29 \( 1 - 55.8iT - 841T^{2} \)
31 \( 1 + 43.4T + 961T^{2} \)
37 \( 1 - 31.5iT - 1.36e3T^{2} \)
41 \( 1 + 51.3iT - 1.68e3T^{2} \)
43 \( 1 + 51.2iT - 1.84e3T^{2} \)
47 \( 1 - 61.7T + 2.20e3T^{2} \)
53 \( 1 - 82.7T + 2.80e3T^{2} \)
59 \( 1 + 97.6iT - 3.48e3T^{2} \)
61 \( 1 - 4.13T + 3.72e3T^{2} \)
67 \( 1 - 63.1iT - 4.48e3T^{2} \)
71 \( 1 + 40.3iT - 5.04e3T^{2} \)
73 \( 1 + 78.5iT - 5.32e3T^{2} \)
79 \( 1 - 51.0T + 6.24e3T^{2} \)
83 \( 1 + 2.72T + 6.88e3T^{2} \)
89 \( 1 + 70.4iT - 7.92e3T^{2} \)
97 \( 1 + 3.44iT - 9.40e3T^{2} \)
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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

−12.26682438543095991587974360770, −10.79018480794268920447149459922, −10.30868024983771027229275003889, −9.202184605365891543986011010523, −8.672013500576773299173348102537, −7.16506028918208060047457370142, −5.62508024232739430740671400735, −5.14564139833729948702269461815, −3.36138716069819525911318356681, −2.22459944930654995620920580826, 1.04354724448697656679772341612, 2.41405479214756510047495721177, 4.07220006359574597880419048412, 5.72698640910203689600564491060, 6.63472719894424126920864440242, 7.55181679535619273472125215136, 8.764595046248849325071486426857, 9.532894075290828571776707305140, 10.77296557733826170712609209240, 11.68669580382662050913607033557

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