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Properties

Label 2-720-240.179-c1-0-2
Degree $2$
Conductor $720$
Sign $-0.202 - 0.979i$
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  + (−1.31 − 0.532i)2-s + (1.43 + 1.39i)4-s + (−1.50 − 1.65i)5-s − 4.30i·7-s + (−1.13 − 2.59i)8-s + (1.09 + 2.96i)10-s + (−4.26 + 4.26i)11-s + (−2.10 + 2.10i)13-s + (−2.29 + 5.64i)14-s + (0.107 + 3.99i)16-s − 3.59·17-s + (0.954 − 0.954i)19-s + (0.144 − 4.46i)20-s + (7.86 − 3.31i)22-s + 6.53·23-s + ⋯
L(s)  = 1  + (−0.926 − 0.376i)2-s + (0.716 + 0.697i)4-s + (−0.673 − 0.738i)5-s − 1.62i·7-s + (−0.401 − 0.915i)8-s + (0.346 + 0.938i)10-s + (−1.28 + 1.28i)11-s + (−0.585 + 0.585i)13-s + (−0.612 + 1.50i)14-s + (0.0269 + 0.999i)16-s − 0.871·17-s + (0.219 − 0.219i)19-s + (0.0323 − 0.999i)20-s + (1.67 − 0.707i)22-s + 1.36·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.202 - 0.979i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0849382 + 0.104294i\)
\(L(\frac12)\) \(\approx\) \(0.0849382 + 0.104294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 + 0.532i)T \)
3 \( 1 \)
5 \( 1 + (1.50 + 1.65i)T \)
good7 \( 1 + 4.30iT - 7T^{2} \)
11 \( 1 + (4.26 - 4.26i)T - 11iT^{2} \)
13 \( 1 + (2.10 - 2.10i)T - 13iT^{2} \)
17 \( 1 + 3.59T + 17T^{2} \)
19 \( 1 + (-0.954 + 0.954i)T - 19iT^{2} \)
23 \( 1 - 6.53T + 23T^{2} \)
29 \( 1 + (-1.84 + 1.84i)T - 29iT^{2} \)
31 \( 1 - 7.64iT - 31T^{2} \)
37 \( 1 + (1.90 + 1.90i)T + 37iT^{2} \)
41 \( 1 - 7.67T + 41T^{2} \)
43 \( 1 + (5.43 - 5.43i)T - 43iT^{2} \)
47 \( 1 - 3.24iT - 47T^{2} \)
53 \( 1 + (6.08 - 6.08i)T - 53iT^{2} \)
59 \( 1 + (2.97 - 2.97i)T - 59iT^{2} \)
61 \( 1 + (0.157 + 0.157i)T + 61iT^{2} \)
67 \( 1 + (0.305 + 0.305i)T + 67iT^{2} \)
71 \( 1 + 1.61iT - 71T^{2} \)
73 \( 1 + 6.90T + 73T^{2} \)
79 \( 1 + 5.39iT - 79T^{2} \)
83 \( 1 + (5.82 - 5.82i)T - 83iT^{2} \)
89 \( 1 + 7.74T + 89T^{2} \)
97 \( 1 + 9.34iT - 97T^{2} \)
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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

−10.65782245454174314926810499938, −9.804201929073189890325487017522, −9.049404051788405035951777757651, −7.980506143967813755151266053676, −7.32096226267821195478756834014, −6.87973495516150289297531858819, −4.82816372779786757512025497532, −4.28471274855205366160175377717, −2.88026866792820366531311320894, −1.35778249482552404919186006605, 0.095695240463062549381021952963, 2.46869149586755178787654351454, 3.02207531177822349068510456010, 5.12701776290142806365458322983, 5.80327024948383569308507215439, 6.75907882320908257279958339379, 7.81356968622369442736037169061, 8.378352631510142795833773154206, 9.110284875366658145486146959015, 10.16106947847003915820567475259

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