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Properties

Label 2-720-144.11-c1-0-10
Degree $2$
Conductor $720$
Sign $0.990 - 0.140i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
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  + (0.497 − 1.32i)2-s + (−1.08 − 1.34i)3-s + (−1.50 − 1.31i)4-s + (−0.965 − 0.258i)5-s + (−2.32 + 0.771i)6-s + (−1.95 + 3.39i)7-s + (−2.49 + 1.33i)8-s + (−0.629 + 2.93i)9-s + (−0.822 + 1.15i)10-s + (4.49 − 1.20i)11-s + (−0.133 + 3.46i)12-s + (−0.501 − 0.134i)13-s + (3.52 + 4.28i)14-s + (0.702 + 1.58i)15-s + (0.535 + 3.96i)16-s + 5.09i·17-s + ⋯
L(s)  = 1  + (0.351 − 0.936i)2-s + (−0.628 − 0.777i)3-s + (−0.752 − 0.658i)4-s + (−0.431 − 0.115i)5-s + (−0.949 + 0.315i)6-s + (−0.740 + 1.28i)7-s + (−0.880 + 0.473i)8-s + (−0.209 + 0.977i)9-s + (−0.260 + 0.363i)10-s + (1.35 − 0.363i)11-s + (−0.0385 + 0.999i)12-s + (−0.139 − 0.0372i)13-s + (0.940 + 1.14i)14-s + (0.181 + 0.408i)15-s + (0.133 + 0.990i)16-s + 1.23i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.990 - 0.140i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.990 - 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.704359 + 0.0497553i\)
\(L(\frac12)\) \(\approx\) \(0.704359 + 0.0497553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.497 + 1.32i)T \)
3 \( 1 + (1.08 + 1.34i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
good7 \( 1 + (1.95 - 3.39i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.49 + 1.20i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.501 + 0.134i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 5.09iT - 17T^{2} \)
19 \( 1 + (2.04 + 2.04i)T + 19iT^{2} \)
23 \( 1 + (-1.14 + 0.659i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.24 - 1.40i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (0.445 - 0.256i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.69 + 1.69i)T + 37iT^{2} \)
41 \( 1 + (-4.41 - 7.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.83 - 6.84i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.84 - 6.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.55 - 7.55i)T - 53iT^{2} \)
59 \( 1 + (-1.89 + 7.09i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.55 - 5.79i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.71 + 10.1i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.02iT - 71T^{2} \)
73 \( 1 - 12.2iT - 73T^{2} \)
79 \( 1 + (-0.100 - 0.0577i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.95 - 14.7i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 4.49T + 89T^{2} \)
97 \( 1 + (2.35 - 4.07i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.87334069873736568749336869352, −9.518386681142679992436375473182, −8.933408190424532756423826151949, −8.002146427645059433672821193536, −6.40375912192222181362172014701, −6.12114701166090989141416599761, −4.97865286920852732107641461420, −3.79294304014367638503603700215, −2.61548109700446168950539714484, −1.37528396583528577919343062775, 0.38517256759295137956844589242, 3.52076211025586860963913815111, 3.97160323262923010756294965700, 4.87073263856500604455341799698, 6.02729660455869729941981997273, 6.94139186600593253596832547403, 7.29996857892211789575525227682, 8.774362678758281829532948764469, 9.536147657856865092731775002800, 10.18639556194834709754672778720

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