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Properties

Label 2-720-80.69-c1-0-31
Degree $2$
Conductor $720$
Sign $0.990 - 0.134i$
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  + (1.40 − 0.112i)2-s + (1.97 − 0.316i)4-s + (−2.18 + 0.466i)5-s + 1.00·7-s + (2.74 − 0.667i)8-s + (−3.03 + 0.903i)10-s + (1.89 + 1.89i)11-s + (2.65 + 2.65i)13-s + (1.40 − 0.112i)14-s + (3.80 − 1.24i)16-s + 1.73i·17-s + (5.33 − 5.33i)19-s + (−4.17 + 1.61i)20-s + (2.88 + 2.45i)22-s − 0.160·23-s + ⋯
L(s)  = 1  + (0.996 − 0.0792i)2-s + (0.987 − 0.158i)4-s + (−0.977 + 0.208i)5-s + 0.378·7-s + (0.971 − 0.235i)8-s + (−0.958 + 0.285i)10-s + (0.571 + 0.571i)11-s + (0.737 + 0.737i)13-s + (0.376 − 0.0299i)14-s + (0.950 − 0.312i)16-s + 0.421i·17-s + (1.22 − 1.22i)19-s + (−0.932 + 0.360i)20-s + (0.614 + 0.524i)22-s − 0.0334·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.990 - 0.134i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.990 - 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.73868 + 0.185265i\)
\(L(\frac12)\) \(\approx\) \(2.73868 + 0.185265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.112i)T \)
3 \( 1 \)
5 \( 1 + (2.18 - 0.466i)T \)
good7 \( 1 - 1.00T + 7T^{2} \)
11 \( 1 + (-1.89 - 1.89i)T + 11iT^{2} \)
13 \( 1 + (-2.65 - 2.65i)T + 13iT^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
19 \( 1 + (-5.33 + 5.33i)T - 19iT^{2} \)
23 \( 1 + 0.160T + 23T^{2} \)
29 \( 1 + (2.70 - 2.70i)T - 29iT^{2} \)
31 \( 1 + 4.64T + 31T^{2} \)
37 \( 1 + (-5.35 + 5.35i)T - 37iT^{2} \)
41 \( 1 + 9.89iT - 41T^{2} \)
43 \( 1 + (7.23 - 7.23i)T - 43iT^{2} \)
47 \( 1 - 4.79iT - 47T^{2} \)
53 \( 1 + (3.44 - 3.44i)T - 53iT^{2} \)
59 \( 1 + (0.101 + 0.101i)T + 59iT^{2} \)
61 \( 1 + (6.01 - 6.01i)T - 61iT^{2} \)
67 \( 1 + (-9.04 - 9.04i)T + 67iT^{2} \)
71 \( 1 + 4.60iT - 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 5.73T + 79T^{2} \)
83 \( 1 + (2.04 + 2.04i)T + 83iT^{2} \)
89 \( 1 + 15.0iT - 89T^{2} \)
97 \( 1 + 3.84iT - 97T^{2} \)
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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

−11.00720613598875711389480361334, −9.634773042108391263063056792260, −8.633330091219513869716287819662, −7.42852950540988902722320816856, −7.01794398576749101314593510148, −5.89249605716956193657595795025, −4.72631052523084270222095865635, −4.03086682404993034914176569177, −3.06282762064762928663835972204, −1.54118266093905306944235028752, 1.33771878705320292523327577761, 3.21464803256148302301034849215, 3.77318521755532359081465124204, 4.92645532729498318119044757232, 5.76297090108261491871687686942, 6.77306879153231760835132979819, 7.889706585592441998640732285049, 8.207157332023130367951804763738, 9.615630207685895371865346157122, 10.76247877107416447055384405350

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