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Properties

Label 2-720-144.11-c1-0-33
Degree $2$
Conductor $720$
Sign $-0.109 - 0.993i$
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  + (0.448 + 1.34i)2-s + (−1.72 + 0.149i)3-s + (−1.59 + 1.20i)4-s + (0.965 + 0.258i)5-s + (−0.973 − 2.24i)6-s + (0.558 − 0.968i)7-s + (−2.32 − 1.60i)8-s + (2.95 − 0.515i)9-s + (0.0855 + 1.41i)10-s + (5.97 − 1.60i)11-s + (2.57 − 2.31i)12-s + (−2.19 − 0.587i)13-s + (1.54 + 0.316i)14-s + (−1.70 − 0.302i)15-s + (1.11 − 3.84i)16-s + 6.56i·17-s + ⋯
L(s)  = 1  + (0.316 + 0.948i)2-s + (−0.996 + 0.0863i)3-s + (−0.799 + 0.600i)4-s + (0.431 + 0.115i)5-s + (−0.397 − 0.917i)6-s + (0.211 − 0.365i)7-s + (−0.823 − 0.567i)8-s + (0.985 − 0.171i)9-s + (0.0270 + 0.446i)10-s + (1.80 − 0.482i)11-s + (0.744 − 0.667i)12-s + (−0.607 − 0.162i)13-s + (0.414 + 0.0844i)14-s + (−0.440 − 0.0780i)15-s + (0.277 − 0.960i)16-s + 1.59i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.109 - 0.993i$
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: \((2,\ 720,\ (\ :1/2),\ -0.109 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.929674 + 1.03759i\)
\(L(\frac12)\) \(\approx\) \(0.929674 + 1.03759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.448 - 1.34i)T \)
3 \( 1 + (1.72 - 0.149i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
good7 \( 1 + (-0.558 + 0.968i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.97 + 1.60i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.19 + 0.587i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 6.56iT - 17T^{2} \)
19 \( 1 + (-3.18 - 3.18i)T + 19iT^{2} \)
23 \( 1 + (-5.38 + 3.10i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.75 - 1.81i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-3.07 + 1.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.91 + 3.91i)T + 37iT^{2} \)
41 \( 1 + (0.657 + 1.13i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.51 - 9.36i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.65 - 4.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.17 + 2.17i)T - 53iT^{2} \)
59 \( 1 + (-0.796 + 2.97i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.37 - 12.5i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.884 + 3.30i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.77iT - 71T^{2} \)
73 \( 1 - 3.66iT - 73T^{2} \)
79 \( 1 + (-13.1 - 7.61i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.511 + 1.90i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + (-2.16 + 3.75i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.65429179658592755575070147095, −9.665878997891148454493970344391, −8.955978340837817180208231464734, −7.77808697651187068654182348590, −6.85688859165621595316573138905, −6.19945018504457605606914750094, −5.49981356721172818866709363182, −4.38102866842179960719155639072, −3.60871340434110920044153804927, −1.24084240367593239797018882506, 0.954588713354924618751230939238, 2.13720785420821833863877746016, 3.65520330650036297818450427474, 4.92876907521647827831772734738, 5.24657683230409070999534362553, 6.54932340633117506950691856432, 7.25887756033620365533699292857, 9.146366300462006829764630904660, 9.343538935025538454616627591100, 10.24707643934789213874209623702

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