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Properties

Label 2-240-15.14-c8-0-20
Degree $2$
Conductor $240$
Sign $-0.818 + 0.573i$
Analytic cond. $97.7708$
Root an. cond. $9.88791$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−58.3 + 56.1i)3-s + (617. + 96.4i)5-s + 3.50e3i·7-s + (252. − 6.55e3i)9-s + 2.07e4i·11-s + 2.12e4i·13-s + (−4.14e4 + 2.90e4i)15-s − 1.40e5·17-s + 1.82e5·19-s + (−1.97e5 − 2.04e5i)21-s − 4.06e5·23-s + (3.72e5 + 1.19e5i)25-s + (3.53e5 + 3.96e5i)27-s − 8.58e4i·29-s − 4.69e5·31-s + ⋯
L(s)  = 1  + (−0.720 + 0.693i)3-s + (0.988 + 0.154i)5-s + 1.46i·7-s + (0.0385 − 0.999i)9-s + 1.41i·11-s + 0.742i·13-s + (−0.818 + 0.573i)15-s − 1.68·17-s + 1.40·19-s + (−1.01 − 1.05i)21-s − 1.45·23-s + (0.952 + 0.304i)25-s + (0.665 + 0.746i)27-s − 0.121i·29-s − 0.508·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.573i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.818 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.818 + 0.573i$
Analytic conductor: \(97.7708\)
Root analytic conductor: \(9.88791\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :4),\ -0.818 + 0.573i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.271153374\)
\(L(\frac12)\) \(\approx\) \(1.271153374\)
\(L(5)\) 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 \( 1 + (58.3 - 56.1i)T \)
5 \( 1 + (-617. - 96.4i)T \)
good7 \( 1 - 3.50e3iT - 5.76e6T^{2} \)
11 \( 1 - 2.07e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.12e4iT - 8.15e8T^{2} \)
17 \( 1 + 1.40e5T + 6.97e9T^{2} \)
19 \( 1 - 1.82e5T + 1.69e10T^{2} \)
23 \( 1 + 4.06e5T + 7.83e10T^{2} \)
29 \( 1 + 8.58e4iT - 5.00e11T^{2} \)
31 \( 1 + 4.69e5T + 8.52e11T^{2} \)
37 \( 1 - 2.30e6iT - 3.51e12T^{2} \)
41 \( 1 - 1.66e6iT - 7.98e12T^{2} \)
43 \( 1 - 1.85e6iT - 1.16e13T^{2} \)
47 \( 1 - 5.76e6T + 2.38e13T^{2} \)
53 \( 1 - 5.45e6T + 6.22e13T^{2} \)
59 \( 1 + 9.28e6iT - 1.46e14T^{2} \)
61 \( 1 - 7.59e6T + 1.91e14T^{2} \)
67 \( 1 - 3.94e7iT - 4.06e14T^{2} \)
71 \( 1 - 1.39e7iT - 6.45e14T^{2} \)
73 \( 1 + 1.62e7iT - 8.06e14T^{2} \)
79 \( 1 + 9.06e6T + 1.51e15T^{2} \)
83 \( 1 - 4.45e7T + 2.25e15T^{2} \)
89 \( 1 + 6.34e6iT - 3.93e15T^{2} \)
97 \( 1 + 1.45e8iT - 7.83e15T^{2} \)
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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

−11.39037235467766345950708356535, −10.04826940203266551020767440364, −9.534253713144754872599343987553, −8.770913478032471418100285745149, −6.96792430251904955679062212480, −6.09880754305924033025075883215, −5.23983812953054330243336865902, −4.32345357701799351694467962184, −2.58845181813499130781001090905, −1.69158990994107751896030078268, 0.33624406804832170313425087422, 0.995141355120817364117470647969, 2.27532220126600863536007260011, 3.84107701763774645285782461867, 5.26564121435480547273065326631, 6.04894610425882727969715109155, 7.01159400701954217974569009879, 7.947721728839037299811904293848, 9.181287100056245098930167834006, 10.56916840589538027181278120538

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