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Properties

Label 2-240-15.14-c2-0-3
Degree $2$
Conductor $240$
Sign $-0.929 - 0.368i$
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  + (−2.23 + 2i)3-s + (2.23 + 4.47i)5-s + 8i·7-s + (1.00 − 8.94i)9-s − 8.94i·11-s + 12i·13-s + (−13.9 − 5.52i)15-s − 31.3·17-s + 6·19-s + (−16 − 17.8i)21-s − 4.47·23-s + (−15.0 + 20.0i)25-s + (15.6 + 22.0i)27-s + 26.8i·29-s − 34·31-s + ⋯
L(s)  = 1  + (−0.745 + 0.666i)3-s + (0.447 + 0.894i)5-s + 1.14i·7-s + (0.111 − 0.993i)9-s − 0.813i·11-s + 0.923i·13-s + (−0.929 − 0.368i)15-s − 1.84·17-s + 0.315·19-s + (−0.761 − 0.851i)21-s − 0.194·23-s + (−0.600 + 0.800i)25-s + (0.579 + 0.814i)27-s + 0.925i·29-s − 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.929 - 0.368i$
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: \((2,\ 240,\ (\ :1),\ -0.929 - 0.368i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.166286 + 0.870688i\)
\(L(\frac12)\) \(\approx\) \(0.166286 + 0.870688i\)
\(L(2)\) 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 \( 1 + (2.23 - 2i)T \)
5 \( 1 + (-2.23 - 4.47i)T \)
good7 \( 1 - 8iT - 49T^{2} \)
11 \( 1 + 8.94iT - 121T^{2} \)
13 \( 1 - 12iT - 169T^{2} \)
17 \( 1 + 31.3T + 289T^{2} \)
19 \( 1 - 6T + 361T^{2} \)
23 \( 1 + 4.47T + 529T^{2} \)
29 \( 1 - 26.8iT - 841T^{2} \)
31 \( 1 + 34T + 961T^{2} \)
37 \( 1 + 44iT - 1.36e3T^{2} \)
41 \( 1 + 17.8iT - 1.68e3T^{2} \)
43 \( 1 - 28iT - 1.84e3T^{2} \)
47 \( 1 + 4.47T + 2.20e3T^{2} \)
53 \( 1 - 40.2T + 2.80e3T^{2} \)
59 \( 1 - 98.3iT - 3.48e3T^{2} \)
61 \( 1 - 74T + 3.72e3T^{2} \)
67 \( 1 + 92iT - 4.48e3T^{2} \)
71 \( 1 - 53.6iT - 5.04e3T^{2} \)
73 \( 1 - 56iT - 5.32e3T^{2} \)
79 \( 1 - 78T + 6.24e3T^{2} \)
83 \( 1 - 102.T + 6.88e3T^{2} \)
89 \( 1 - 17.8iT - 7.92e3T^{2} \)
97 \( 1 - 32iT - 9.40e3T^{2} \)
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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

−12.02091417594149571467253106969, −11.20638278571697718906696487346, −10.68412491441791367292374240553, −9.324636149143746079768920060898, −8.875906821821226134083505891165, −6.97291429084686988208839233027, −6.14750962579094369219611751526, −5.27800358351991013344095605343, −3.80620736054263723308705339874, −2.31988866468879359275629985219, 0.49902437007466475314303390076, 1.96939373374462807014006082662, 4.29089109209537392358797095286, 5.20755201203946116914070106762, 6.44437927171065323148127677280, 7.37274004142461945432821216038, 8.373462495102438507997040234245, 9.729458916074137899546151610736, 10.57175543853368695288055309376, 11.52965891341198642680207422659

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