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Properties

Label 2-240-16.13-c1-0-1
Degree $2$
Conductor $240$
Sign $0.828 - 0.560i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.04 − 0.948i)2-s + (−0.707 − 0.707i)3-s + (0.202 + 1.98i)4-s + (−0.707 + 0.707i)5-s + (0.0715 + 1.41i)6-s + 0.740i·7-s + (1.67 − 2.27i)8-s + 1.00i·9-s + (1.41 − 0.0715i)10-s + (−3.83 + 3.83i)11-s + (1.26 − 1.54i)12-s + (3.31 + 3.31i)13-s + (0.701 − 0.776i)14-s + 1.00·15-s + (−3.91 + 0.804i)16-s + 2.93·17-s + ⋯
L(s)  = 1  + (−0.741 − 0.670i)2-s + (−0.408 − 0.408i)3-s + (0.101 + 0.994i)4-s + (−0.316 + 0.316i)5-s + (0.0292 + 0.576i)6-s + 0.279i·7-s + (0.592 − 0.805i)8-s + 0.333i·9-s + (0.446 − 0.0226i)10-s + (−1.15 + 1.15i)11-s + (0.364 − 0.447i)12-s + (0.918 + 0.918i)13-s + (0.187 − 0.207i)14-s + 0.258·15-s + (−0.979 + 0.201i)16-s + 0.712·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.828 - 0.560i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.828 - 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.588358 + 0.180445i\)
\(L(\frac12)\) \(\approx\) \(0.588358 + 0.180445i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.04 + 0.948i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - 0.740iT - 7T^{2} \)
11 \( 1 + (3.83 - 3.83i)T - 11iT^{2} \)
13 \( 1 + (-3.31 - 3.31i)T + 13iT^{2} \)
17 \( 1 - 2.93T + 17T^{2} \)
19 \( 1 + (-5.02 - 5.02i)T + 19iT^{2} \)
23 \( 1 + 5.45iT - 23T^{2} \)
29 \( 1 + (-2.64 - 2.64i)T + 29iT^{2} \)
31 \( 1 + 5.94T + 31T^{2} \)
37 \( 1 + (0.479 - 0.479i)T - 37iT^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + (4.93 - 4.93i)T - 43iT^{2} \)
47 \( 1 + 8.15T + 47T^{2} \)
53 \( 1 + (5.05 - 5.05i)T - 53iT^{2} \)
59 \( 1 + (-3.83 + 3.83i)T - 59iT^{2} \)
61 \( 1 + (4.87 + 4.87i)T + 61iT^{2} \)
67 \( 1 + (3.99 + 3.99i)T + 67iT^{2} \)
71 \( 1 + 3.55iT - 71T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + (-4.61 - 4.61i)T + 83iT^{2} \)
89 \( 1 - 2.62iT - 89T^{2} \)
97 \( 1 - 1.67T + 97T^{2} \)
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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

−12.15823379923924015691390475774, −11.23635663446197838221094385841, −10.37592043734416452575121184542, −9.528243720289993514313028692857, −8.191737168536671428815842252550, −7.51727147815802070042302875249, −6.39927480341886287529751138712, −4.78186592853828819165885420822, −3.23097589681402420624990044320, −1.71285220943118071034055967252, 0.69930547686957126613325545926, 3.38234572981551460275669733707, 5.25141773041129576527981791347, 5.70097560954561912936275345089, 7.24765568116188341716692653664, 8.109827109252861771443499204793, 9.007836162211549297940964808332, 10.14393420010203962738242498393, 10.88749669515982192117209993106, 11.63848963740168155126494381462

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