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Properties

Label 2-720-144.11-c1-0-3
Degree $2$
Conductor $720$
Sign $0.621 - 0.783i$
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  + (−0.342 − 1.37i)2-s + (0.254 − 1.71i)3-s + (−1.76 + 0.939i)4-s + (−0.965 − 0.258i)5-s + (−2.43 + 0.236i)6-s + (−0.158 + 0.273i)7-s + (1.89 + 2.10i)8-s + (−2.87 − 0.872i)9-s + (−0.0246 + 1.41i)10-s + (−3.26 + 0.876i)11-s + (1.15 + 3.26i)12-s + (−1.09 − 0.293i)13-s + (0.430 + 0.123i)14-s + (−0.689 + 1.58i)15-s + (2.23 − 3.31i)16-s + 4.42i·17-s + ⋯
L(s)  = 1  + (−0.241 − 0.970i)2-s + (0.147 − 0.989i)3-s + (−0.882 + 0.469i)4-s + (−0.431 − 0.115i)5-s + (−0.995 + 0.0966i)6-s + (−0.0597 + 0.103i)7-s + (0.669 + 0.743i)8-s + (−0.956 − 0.290i)9-s + (−0.00778 + 0.447i)10-s + (−0.985 + 0.264i)11-s + (0.334 + 0.942i)12-s + (−0.303 − 0.0812i)13-s + (0.114 + 0.0329i)14-s + (−0.178 + 0.410i)15-s + (0.559 − 0.829i)16-s + 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.621 - 0.783i$
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: \((2,\ 720,\ (\ :1/2),\ 0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.172905 + 0.0835752i\)
\(L(\frac12)\) \(\approx\) \(0.172905 + 0.0835752i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 + 1.37i)T \)
3 \( 1 + (-0.254 + 1.71i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
good7 \( 1 + (0.158 - 0.273i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.26 - 0.876i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.09 + 0.293i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 4.42iT - 17T^{2} \)
19 \( 1 + (-1.01 - 1.01i)T + 19iT^{2} \)
23 \( 1 + (3.26 - 1.88i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.83 - 0.490i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (0.144 - 0.0835i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.92 - 4.92i)T + 37iT^{2} \)
41 \( 1 + (1.56 + 2.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.65 + 9.89i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.15 - 7.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.29 - 7.29i)T - 53iT^{2} \)
59 \( 1 + (-3.00 + 11.2i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.932 + 3.47i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.31 - 4.89i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.833iT - 71T^{2} \)
73 \( 1 + 4.99iT - 73T^{2} \)
79 \( 1 + (4.31 + 2.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.73 + 13.9i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 8.33T + 89T^{2} \)
97 \( 1 + (1.79 - 3.10i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.63745496964583646447427590999, −9.762143960327242978008319902525, −8.741631063303028677509614798557, −7.960301055167468531248046284586, −7.48316683273315990156957452412, −6.07208571600497938086440160885, −5.00110325491366884685740552358, −3.71532781417696654550827018868, −2.66206909431681880520745298208, −1.56863259261124630268742461393, 0.10544122398535483098766633264, 2.81262360311080910323100669216, 4.05288622101395556201382406107, 4.93914419905305144963482570955, 5.66939045121902321001962222945, 6.87309022342021738405622893319, 7.83723466184497574626401804107, 8.433037495881722703596325943731, 9.475180021193603353844147133541, 9.999599328069824787970540880713

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