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Properties

Label 2-240-60.47-c2-0-12
Degree $2$
Conductor $240$
Sign $0.552 - 0.833i$
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  + (0.507 + 2.95i)3-s + (3.53 + 3.53i)5-s + (8.66 − 8.66i)7-s + (−8.48 + 3i)9-s + 17.1·11-s + (4 − 4i)13-s + (−8.66 + 12.2i)15-s + (−15.5 + 15.5i)17-s − 6.92·19-s + (30 + 21.2i)21-s + (−4.89 + 4.89i)23-s + 25.0i·25-s + (−13.1 − 23.5i)27-s − 26.8·29-s + 3.46i·31-s + ⋯
L(s)  = 1  + (0.169 + 0.985i)3-s + (0.707 + 0.707i)5-s + (1.23 − 1.23i)7-s + (−0.942 + 0.333i)9-s + 1.55·11-s + (0.307 − 0.307i)13-s + (−0.577 + 0.816i)15-s + (−0.915 + 0.915i)17-s − 0.364·19-s + (1.42 + 1.01i)21-s + (−0.212 + 0.212i)23-s + 1.00i·25-s + (−0.487 − 0.872i)27-s − 0.926·29-s + 0.111i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.552 - 0.833i$
Analytic conductor: \(6.53952\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1),\ 0.552 - 0.833i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.85328 + 0.994827i\)
\(L(\frac12)\) \(\approx\) \(1.85328 + 0.994827i\)
\(L(2)\) 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 \( 1 + (-0.507 - 2.95i)T \)
5 \( 1 + (-3.53 - 3.53i)T \)
good7 \( 1 + (-8.66 + 8.66i)T - 49iT^{2} \)
11 \( 1 - 17.1T + 121T^{2} \)
13 \( 1 + (-4 + 4i)T - 169iT^{2} \)
17 \( 1 + (15.5 - 15.5i)T - 289iT^{2} \)
19 \( 1 + 6.92T + 361T^{2} \)
23 \( 1 + (4.89 - 4.89i)T - 529iT^{2} \)
29 \( 1 + 26.8T + 841T^{2} \)
31 \( 1 - 3.46iT - 961T^{2} \)
37 \( 1 + (-38 - 38i)T + 1.36e3iT^{2} \)
41 \( 1 + 62.2iT - 1.68e3T^{2} \)
43 \( 1 + (24.2 + 24.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (-9.79 - 9.79i)T + 2.20e3iT^{2} \)
53 \( 1 + (-5.65 - 5.65i)T + 2.80e3iT^{2} \)
59 \( 1 + 46.5iT - 3.48e3T^{2} \)
61 \( 1 + 42T + 3.72e3T^{2} \)
67 \( 1 + (20.7 - 20.7i)T - 4.48e3iT^{2} \)
71 \( 1 - 14.6T + 5.04e3T^{2} \)
73 \( 1 + (37 - 37i)T - 5.32e3iT^{2} \)
79 \( 1 + 72.7T + 6.24e3T^{2} \)
83 \( 1 + (-93.0 + 93.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 87.6T + 7.92e3T^{2} \)
97 \( 1 + (-35 - 35i)T + 9.40e3iT^{2} \)
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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

−11.58926016367878845628757501148, −10.91774484924934028897849488743, −10.32285573558262122371273588307, −9.255411763709924900556078095328, −8.283704776201467930337219034304, −6.99848409717742811991376665005, −5.88133385274371180716831105982, −4.44740881711119796094395417754, −3.67412998109568464132937984862, −1.75228934178341575662868127606, 1.40121457065310230003606169105, 2.32742728383995980215461133726, 4.51356168908369619064333736546, 5.74093074698701479101860062294, 6.55442060438327508096419044792, 7.961835912787694581852853731671, 8.964327601669516973132388301598, 9.230323873293411655880922486123, 11.31996486585850488105875095871, 11.73429216764948338047999727185

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