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Properties

Label 2-720-144.11-c1-0-30
Degree $2$
Conductor $720$
Sign $0.617 - 0.786i$
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  + (0.286 + 1.38i)2-s + (0.113 − 1.72i)3-s + (−1.83 + 0.794i)4-s + (−0.965 − 0.258i)5-s + (2.42 − 0.339i)6-s + (−0.903 + 1.56i)7-s + (−1.62 − 2.31i)8-s + (−2.97 − 0.390i)9-s + (0.0813 − 1.41i)10-s + (1.94 − 0.522i)11-s + (1.16 + 3.26i)12-s + (6.02 + 1.61i)13-s + (−2.42 − 0.802i)14-s + (−0.556 + 1.64i)15-s + (2.73 − 2.91i)16-s + 3.29i·17-s + ⋯
L(s)  = 1  + (0.202 + 0.979i)2-s + (0.0652 − 0.997i)3-s + (−0.917 + 0.397i)4-s + (−0.431 − 0.115i)5-s + (0.990 − 0.138i)6-s + (−0.341 + 0.591i)7-s + (−0.575 − 0.818i)8-s + (−0.991 − 0.130i)9-s + (0.0257 − 0.446i)10-s + (0.587 − 0.157i)11-s + (0.336 + 0.941i)12-s + (1.67 + 0.447i)13-s + (−0.648 − 0.214i)14-s + (−0.143 + 0.423i)15-s + (0.684 − 0.729i)16-s + 0.797i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.617 - 0.786i$
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: \((2,\ 720,\ (\ :1/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29486 + 0.629216i\)
\(L(\frac12)\) \(\approx\) \(1.29486 + 0.629216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.286 - 1.38i)T \)
3 \( 1 + (-0.113 + 1.72i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
good7 \( 1 + (0.903 - 1.56i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.94 + 0.522i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-6.02 - 1.61i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 3.29iT - 17T^{2} \)
19 \( 1 + (-2.61 - 2.61i)T + 19iT^{2} \)
23 \( 1 + (-5.31 + 3.06i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.04 + 2.15i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (4.33 - 2.50i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.969 + 0.969i)T + 37iT^{2} \)
41 \( 1 + (-1.60 - 2.77i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.831 + 3.10i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-5.72 + 9.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.14 - 5.14i)T - 53iT^{2} \)
59 \( 1 + (2.10 - 7.87i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.412 + 1.54i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.65 - 6.15i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.83iT - 71T^{2} \)
73 \( 1 - 12.2iT - 73T^{2} \)
79 \( 1 + (-6.52 - 3.76i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.40 + 16.4i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 0.785T + 89T^{2} \)
97 \( 1 + (-1.71 + 2.97i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.58209053796461529288763801631, −8.980827733113892717988213855151, −8.759705910207232615168669231470, −7.915502717572757102476472581777, −6.86326748310587834958986386678, −6.26670765286123872480010586791, −5.53576217775161713181244134018, −4.08353827876219023260097689970, −3.11999375617143308429081504466, −1.16823828328585197252889705144, 0.931368668968438129054779028202, 3.01375076955579270959762067489, 3.58745461932409258180419086745, 4.50411549363768000290076892379, 5.43355518267983479963879276465, 6.61797143466185715115136383053, 7.996062757094175981483547065239, 9.014965245856380696298289042795, 9.462308742713445775445822611821, 10.50789166101106322679354422555

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