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Properties

Label 2-240-80.29-c1-0-23
Degree $2$
Conductor $240$
Sign $-0.880 - 0.473i$
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  + (0.550 − 1.30i)2-s + (−0.707 + 0.707i)3-s + (−1.39 − 1.43i)4-s + (−2.23 − 0.162i)5-s + (0.531 + 1.31i)6-s − 2.93·7-s + (−2.63 + 1.02i)8-s − 1.00i·9-s + (−1.43 + 2.81i)10-s + (−0.663 + 0.663i)11-s + (1.99 + 0.0281i)12-s + (−1.12 + 1.12i)13-s + (−1.61 + 3.82i)14-s + (1.69 − 1.46i)15-s + (−0.112 + 3.99i)16-s − 7.47i·17-s + ⋯
L(s)  = 1  + (0.389 − 0.921i)2-s + (−0.408 + 0.408i)3-s + (−0.697 − 0.716i)4-s + (−0.997 − 0.0724i)5-s + (0.217 + 0.534i)6-s − 1.10·7-s + (−0.931 + 0.363i)8-s − 0.333i·9-s + (−0.454 + 0.890i)10-s + (−0.200 + 0.200i)11-s + (0.577 + 0.00813i)12-s + (−0.312 + 0.312i)13-s + (−0.431 + 1.02i)14-s + (0.436 − 0.377i)15-s + (−0.0281 + 0.999i)16-s − 1.81i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.880 - 0.473i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.880 - 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0595753 + 0.236579i\)
\(L(\frac12)\) \(\approx\) \(0.0595753 + 0.236579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.550 + 1.30i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (2.23 + 0.162i)T \)
good7 \( 1 + 2.93T + 7T^{2} \)
11 \( 1 + (0.663 - 0.663i)T - 11iT^{2} \)
13 \( 1 + (1.12 - 1.12i)T - 13iT^{2} \)
17 \( 1 + 7.47iT - 17T^{2} \)
19 \( 1 + (-0.423 - 0.423i)T + 19iT^{2} \)
23 \( 1 + 6.17T + 23T^{2} \)
29 \( 1 + (2.95 + 2.95i)T + 29iT^{2} \)
31 \( 1 - 1.82T + 31T^{2} \)
37 \( 1 + (-5.53 - 5.53i)T + 37iT^{2} \)
41 \( 1 + 12.3iT - 41T^{2} \)
43 \( 1 + (0.897 + 0.897i)T + 43iT^{2} \)
47 \( 1 - 4.12iT - 47T^{2} \)
53 \( 1 + (0.146 + 0.146i)T + 53iT^{2} \)
59 \( 1 + (7.72 - 7.72i)T - 59iT^{2} \)
61 \( 1 + (7.37 + 7.37i)T + 61iT^{2} \)
67 \( 1 + (8.68 - 8.68i)T - 67iT^{2} \)
71 \( 1 + 8.95iT - 71T^{2} \)
73 \( 1 + 0.174T + 73T^{2} \)
79 \( 1 - 3.06T + 79T^{2} \)
83 \( 1 + (-9.18 + 9.18i)T - 83iT^{2} \)
89 \( 1 - 8.71iT - 89T^{2} \)
97 \( 1 + 10.5iT - 97T^{2} \)
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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

−11.88063009876056206768523487816, −10.73470979409484621785326784235, −9.781681264532001542739053034193, −9.122194921828709242482965152541, −7.56680092283685938442323330711, −6.25750144158304550441092791906, −4.94673014370976247602460892062, −3.97031117218741798811293508369, −2.84229838731846964480311450585, −0.17598363074291378742856037647, 3.25746160171060681141419038299, 4.33423443694604933958647262991, 5.84052868240819814641312971617, 6.55385155306543043992890304425, 7.67542439317920147730261994098, 8.333900138997615297472624555609, 9.685410660568204565998658924269, 10.89256454934910389014014806338, 12.15638407803474793591257114024, 12.68387477703780253756272538553

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