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Properties

Label 2-720-144.11-c1-0-1
Degree $2$
Conductor $720$
Sign $-0.835 + 0.549i$
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  + (0.130 + 1.40i)2-s + (0.00582 − 1.73i)3-s + (−1.96 + 0.367i)4-s + (0.965 + 0.258i)5-s + (2.43 − 0.217i)6-s + (−1.28 + 2.22i)7-s + (−0.774 − 2.72i)8-s + (−2.99 − 0.0201i)9-s + (−0.238 + 1.39i)10-s + (−1.49 + 0.400i)11-s + (0.625 + 3.40i)12-s + (−4.35 − 1.16i)13-s + (−3.30 − 1.52i)14-s + (0.453 − 1.67i)15-s + (3.72 − 1.44i)16-s − 3.11i·17-s + ⋯
L(s)  = 1  + (0.0923 + 0.995i)2-s + (0.00336 − 0.999i)3-s + (−0.982 + 0.183i)4-s + (0.431 + 0.115i)5-s + (0.996 − 0.0889i)6-s + (−0.486 + 0.842i)7-s + (−0.273 − 0.961i)8-s + (−0.999 − 0.00672i)9-s + (−0.0753 + 0.440i)10-s + (−0.451 + 0.120i)11-s + (0.180 + 0.983i)12-s + (−1.20 − 0.323i)13-s + (−0.883 − 0.406i)14-s + (0.117 − 0.431i)15-s + (0.932 − 0.361i)16-s − 0.754i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.835 + 0.549i$
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: \((2,\ 720,\ (\ :1/2),\ -0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00940660 - 0.0314534i\)
\(L(\frac12)\) \(\approx\) \(0.00940660 - 0.0314534i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.130 - 1.40i)T \)
3 \( 1 + (-0.00582 + 1.73i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
good7 \( 1 + (1.28 - 2.22i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.49 - 0.400i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (4.35 + 1.16i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 3.11iT - 17T^{2} \)
19 \( 1 + (-0.356 - 0.356i)T + 19iT^{2} \)
23 \( 1 + (1.88 - 1.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.55 - 2.29i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (3.82 - 2.20i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.60 + 6.60i)T + 37iT^{2} \)
41 \( 1 + (0.181 + 0.315i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.153 + 0.574i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.01 - 5.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.622 + 0.622i)T - 53iT^{2} \)
59 \( 1 + (-1.61 + 6.02i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.62 + 13.5i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.917 + 3.42i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 14.8iT - 71T^{2} \)
73 \( 1 + 9.34iT - 73T^{2} \)
79 \( 1 + (-9.07 - 5.23i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.29 - 12.2i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 1.43T + 89T^{2} \)
97 \( 1 + (4.05 - 7.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.91120173630113558360895929595, −9.576553422327924511987290768798, −9.196032864939841762466347740601, −8.049829385650412957317565327426, −7.34316085782793591663124508397, −6.64415270673930089875724048064, −5.56429213878310479950102084928, −5.20795705724363344152107776057, −3.32763188504648325377424907581, −2.17368790193936033928381371875, 0.01525003720748647956386551917, 2.10344874499992121721149740529, 3.32427697246923842203579617107, 4.17365864933031444062647052025, 5.06923027596488162687057155024, 5.96394123067477398620859153069, 7.42767422268728794398596595299, 8.583043874597592735721880157841, 9.416185838586804786163713356912, 10.11893839866746439125138847191

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