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Properties

Label 2-720-80.27-c1-0-34
Degree $2$
Conductor $720$
Sign $0.994 + 0.109i$
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  + (−1.41 + 0.0660i)2-s + (1.99 − 0.186i)4-s + (2.00 + 0.987i)5-s + (1.55 + 1.55i)7-s + (−2.80 + 0.395i)8-s + (−2.89 − 1.26i)10-s + (4.19 − 4.19i)11-s − 5.09i·13-s + (−2.29 − 2.09i)14-s + (3.93 − 0.743i)16-s + (−0.213 − 0.213i)17-s + (0.844 − 0.844i)19-s + (4.17 + 1.59i)20-s + (−5.65 + 6.20i)22-s + (−1.70 + 1.70i)23-s + ⋯
L(s)  = 1  + (−0.998 + 0.0467i)2-s + (0.995 − 0.0933i)4-s + (0.897 + 0.441i)5-s + (0.587 + 0.587i)7-s + (−0.990 + 0.139i)8-s + (−0.916 − 0.399i)10-s + (1.26 − 1.26i)11-s − 1.41i·13-s + (−0.614 − 0.559i)14-s + (0.982 − 0.185i)16-s + (−0.0517 − 0.0517i)17-s + (0.193 − 0.193i)19-s + (0.934 + 0.355i)20-s + (−1.20 + 1.32i)22-s + (−0.356 + 0.356i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.994 + 0.109i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.994 + 0.109i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30936 - 0.0716696i\)
\(L(\frac12)\) \(\approx\) \(1.30936 - 0.0716696i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0660i)T \)
3 \( 1 \)
5 \( 1 + (-2.00 - 0.987i)T \)
good7 \( 1 + (-1.55 - 1.55i)T + 7iT^{2} \)
11 \( 1 + (-4.19 + 4.19i)T - 11iT^{2} \)
13 \( 1 + 5.09iT - 13T^{2} \)
17 \( 1 + (0.213 + 0.213i)T + 17iT^{2} \)
19 \( 1 + (-0.844 + 0.844i)T - 19iT^{2} \)
23 \( 1 + (1.70 - 1.70i)T - 23iT^{2} \)
29 \( 1 + (2.24 + 2.24i)T + 29iT^{2} \)
31 \( 1 - 0.818iT - 31T^{2} \)
37 \( 1 + 5.12iT - 37T^{2} \)
41 \( 1 + 3.34iT - 41T^{2} \)
43 \( 1 - 4.49iT - 43T^{2} \)
47 \( 1 + (4.29 - 4.29i)T - 47iT^{2} \)
53 \( 1 - 1.00T + 53T^{2} \)
59 \( 1 + (-7.65 - 7.65i)T + 59iT^{2} \)
61 \( 1 + (1.90 - 1.90i)T - 61iT^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + (-2.70 - 2.70i)T + 73iT^{2} \)
79 \( 1 + 8.32T + 79T^{2} \)
83 \( 1 - 9.17T + 83T^{2} \)
89 \( 1 - 4.25T + 89T^{2} \)
97 \( 1 + (7.15 + 7.15i)T + 97iT^{2} \)
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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

−10.30474808880256030147282185307, −9.462376610299948242770265902650, −8.756181145721375670359248324135, −8.017695060068903468028374807983, −6.94554545968938648735580790431, −5.93600939441700331431570284714, −5.52122190035503927956165888244, −3.47402326921719098676618489065, −2.42529080253536476815135331010, −1.12591427189849845333061736068, 1.41911897526617904150632197472, 2.04637133957110228225353267028, 3.95251830844516380229136827870, 4.97444381039582193690597236758, 6.44742527268103945530542419448, 6.85539164516091205784942148808, 7.947516914922277431951186733317, 8.961040580641316782554204608702, 9.522587084734093929269850106949, 10.13290423427782897001673793952

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