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Properties

Label 2-720-45.34-c1-0-8
Degree $2$
Conductor $720$
Sign $-0.262 - 0.964i$
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  + (−1.57 + 0.724i)3-s + (−1.81 + 1.30i)5-s + (3.85 + 2.22i)7-s + (1.94 − 2.28i)9-s + (0.724 − 1.25i)11-s + (2.12 − 1.22i)13-s + (1.90 − 3.37i)15-s + 3.89i·17-s − 0.550·19-s + (−7.67 − 0.707i)21-s + (−2.51 + 1.44i)23-s + (1.58 − 4.74i)25-s + (−1.41 + 5.00i)27-s + (−3 + 5.19i)29-s + (3.22 + 5.58i)31-s + ⋯
L(s)  = 1  + (−0.908 + 0.418i)3-s + (−0.811 + 0.584i)5-s + (1.45 + 0.840i)7-s + (0.649 − 0.760i)9-s + (0.218 − 0.378i)11-s + (0.588 − 0.339i)13-s + (0.492 − 0.870i)15-s + 0.945i·17-s − 0.126·19-s + (−1.67 − 0.154i)21-s + (−0.523 + 0.302i)23-s + (0.316 − 0.948i)25-s + (−0.272 + 0.962i)27-s + (−0.557 + 0.964i)29-s + (0.579 + 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.262 - 0.964i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.262 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.625308 + 0.818457i\)
\(L(\frac12)\) \(\approx\) \(0.625308 + 0.818457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.57 - 0.724i)T \)
5 \( 1 + (1.81 - 1.30i)T \)
good7 \( 1 + (-3.85 - 2.22i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.724 + 1.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.12 + 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.89iT - 17T^{2} \)
19 \( 1 + 0.550T + 19T^{2} \)
23 \( 1 + (2.51 - 1.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.22 - 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.45 + 3.72i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.389 - 0.224i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.44iT - 53T^{2} \)
59 \( 1 + (-5.62 - 9.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.224 + 0.389i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.18 - 4.72i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 - 4.79iT - 73T^{2} \)
79 \( 1 + (-3.67 + 6.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.46 - 2i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + (11.2 + 6.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.79391579150234321915170006786, −10.16196294767204722174441750588, −8.662487563522191844612508237488, −8.274447375871303293248099250257, −7.09392594617220263982216680730, −6.10244122544745394796094422848, −5.26982558831089612806032912066, −4.33056773562902431073772473137, −3.30862509948901840703181187809, −1.50737976413979038467118872780, 0.66898803967955416033878543461, 1.85548504514922724952101685823, 4.10982176542913523214242395981, 4.56231524818238876539020097141, 5.54094547759228836725306246456, 6.76637344140690262571006927088, 7.68197135316107823031510946404, 8.047791899258637901840362570147, 9.287537516404179403019667148460, 10.44033024461097089033218514347

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