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Properties

Label 2-240-80.3-c1-0-12
Degree $2$
Conductor $240$
Sign $0.144 - 0.989i$
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  + (0.812 + 1.15i)2-s + 3-s + (−0.679 + 1.88i)4-s + (1.45 + 1.69i)5-s + (0.812 + 1.15i)6-s + (1.12 − 1.12i)7-s + (−2.72 + 0.741i)8-s + 9-s + (−0.780 + 3.06i)10-s + (−4.05 − 4.05i)11-s + (−0.679 + 1.88i)12-s − 1.51i·13-s + (2.22 + 0.389i)14-s + (1.45 + 1.69i)15-s + (−3.07 − 2.55i)16-s + (−1.61 + 1.61i)17-s + ⋯
L(s)  = 1  + (0.574 + 0.818i)2-s + 0.577·3-s + (−0.339 + 0.940i)4-s + (0.651 + 0.758i)5-s + (0.331 + 0.472i)6-s + (0.426 − 0.426i)7-s + (−0.965 + 0.261i)8-s + 0.333·9-s + (−0.246 + 0.969i)10-s + (−1.22 − 1.22i)11-s + (−0.196 + 0.542i)12-s − 0.419i·13-s + (0.593 + 0.104i)14-s + (0.376 + 0.438i)15-s + (−0.768 − 0.639i)16-s + (−0.392 + 0.392i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.144 - 0.989i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49776 + 1.29451i\)
\(L(\frac12)\) \(\approx\) \(1.49776 + 1.29451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.812 - 1.15i)T \)
3 \( 1 - T \)
5 \( 1 + (-1.45 - 1.69i)T \)
good7 \( 1 + (-1.12 + 1.12i)T - 7iT^{2} \)
11 \( 1 + (4.05 + 4.05i)T + 11iT^{2} \)
13 \( 1 + 1.51iT - 13T^{2} \)
17 \( 1 + (1.61 - 1.61i)T - 17iT^{2} \)
19 \( 1 + (-3.09 - 3.09i)T + 19iT^{2} \)
23 \( 1 + (-1.55 - 1.55i)T + 23iT^{2} \)
29 \( 1 + (-0.425 + 0.425i)T - 29iT^{2} \)
31 \( 1 + 9.02iT - 31T^{2} \)
37 \( 1 + 7.76iT - 37T^{2} \)
41 \( 1 - 7.27iT - 41T^{2} \)
43 \( 1 + 9.30iT - 43T^{2} \)
47 \( 1 + (1.13 + 1.13i)T + 47iT^{2} \)
53 \( 1 + 8.27T + 53T^{2} \)
59 \( 1 + (9.12 - 9.12i)T - 59iT^{2} \)
61 \( 1 + (-4.89 - 4.89i)T + 61iT^{2} \)
67 \( 1 + 2.17iT - 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + (1.11 - 1.11i)T - 73iT^{2} \)
79 \( 1 - 1.68T + 79T^{2} \)
83 \( 1 + 5.64T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + (6.26 - 6.26i)T - 97iT^{2} \)
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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

−12.84053635611002507530391310611, −11.33116260943408193821051577198, −10.46746096810538558349760635425, −9.266733817948888036897627838293, −8.039475412875926918224455627136, −7.52172837440727794520927170264, −6.15582523832372827518086155435, −5.33466290645742384323535971799, −3.73571343025346480482974186546, −2.65463885879279742873597854966, 1.76021528153817121964946328420, 2.83216136779539023217383874721, 4.75932678728650680050524826551, 5.09093230659663919874040761890, 6.75517211830322708099351528655, 8.264052162259283070385838903170, 9.280069589065782706036261312807, 9.911433652731222794298550232855, 10.99457079878686295101082217622, 12.20056864632534812102779180821

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