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Properties

Label 2-240-80.43-c1-0-2
Degree $2$
Conductor $240$
Sign $-0.441 - 0.897i$
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  + (−1.15 − 0.812i)2-s + i·3-s + (0.679 + 1.88i)4-s + (−1.69 + 1.45i)5-s + (0.812 − 1.15i)6-s + (1.12 − 1.12i)7-s + (0.741 − 2.72i)8-s − 9-s + (3.14 − 0.307i)10-s + (−4.05 + 4.05i)11-s + (−1.88 + 0.679i)12-s − 1.51·13-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  + (−0.818 − 0.574i)2-s + 0.577i·3-s + (0.339 + 0.940i)4-s + (−0.758 + 0.651i)5-s + (0.331 − 0.472i)6-s + (0.426 − 0.426i)7-s + (0.261 − 0.965i)8-s − 0.333·9-s + (0.995 − 0.0972i)10-s + (−1.22 + 1.22i)11-s + (−0.542 + 0.196i)12-s − 0.419·13-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.441 - 0.897i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.441 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.246661 + 0.396070i\)
\(L(\frac12)\) \(\approx\) \(0.246661 + 0.396070i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−12.19517273138607953187441132597, −11.16655078742834385592589096058, −10.46298279947620578307336683067, −9.916833312480283499396209483440, −8.496353624563839083255921464194, −7.70697851103188643265628261871, −6.85334758504653699521922375181, −4.80725960023475443895440468965, −3.72459891857074856271933828056, −2.32812403377168553795635315453, 0.46772168344636450156766967672, 2.51089633504180272538736097659, 4.80404653219953504558777123736, 5.77262452178254568027514644487, 7.05018116149443993707758049008, 8.189340055972017372357095204302, 8.398763107605784828949514502847, 9.626785232167452201267818842623, 11.13627746747951707020563577609, 11.42992224537403418042837591383

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