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Properties

Label 2-720-144.11-c1-0-2
Degree $2$
Conductor $720$
Sign $-0.649 - 0.760i$
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  + (0.948 − 1.04i)2-s + (−1.03 + 1.38i)3-s + (−0.198 − 1.99i)4-s + (−0.965 − 0.258i)5-s + (0.473 + 2.40i)6-s + (−0.374 + 0.649i)7-s + (−2.27 − 1.67i)8-s + (−0.857 − 2.87i)9-s + (−1.18 + 0.767i)10-s + (−3.66 + 0.982i)11-s + (2.96 + 1.78i)12-s + (3.15 + 0.844i)13-s + (0.325 + 1.00i)14-s + (1.35 − 1.07i)15-s + (−3.92 + 0.791i)16-s + 6.29i·17-s + ⋯
L(s)  = 1  + (0.671 − 0.741i)2-s + (−0.597 + 0.801i)3-s + (−0.0994 − 0.995i)4-s + (−0.431 − 0.115i)5-s + (0.193 + 0.981i)6-s + (−0.141 + 0.245i)7-s + (−0.804 − 0.593i)8-s + (−0.285 − 0.958i)9-s + (−0.375 + 0.242i)10-s + (−1.10 + 0.296i)11-s + (0.857 + 0.514i)12-s + (0.874 + 0.234i)13-s + (0.0868 + 0.269i)14-s + (0.350 − 0.277i)15-s + (−0.980 + 0.197i)16-s + 1.52i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.649 - 0.760i$
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: \((2,\ 720,\ (\ :1/2),\ -0.649 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0951303 + 0.206451i\)
\(L(\frac12)\) \(\approx\) \(0.0951303 + 0.206451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.948 + 1.04i)T \)
3 \( 1 + (1.03 - 1.38i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
good7 \( 1 + (0.374 - 0.649i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.66 - 0.982i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.15 - 0.844i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 6.29iT - 17T^{2} \)
19 \( 1 + (3.26 + 3.26i)T + 19iT^{2} \)
23 \( 1 + (6.11 - 3.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.82 - 1.29i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (7.57 - 4.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.87 + 4.87i)T + 37iT^{2} \)
41 \( 1 + (-2.10 - 3.64i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.23 + 12.0i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.64 + 4.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.61 + 4.61i)T - 53iT^{2} \)
59 \( 1 + (-0.589 + 2.19i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.523 + 1.95i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.45 - 9.16i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.14iT - 71T^{2} \)
73 \( 1 + 4.18iT - 73T^{2} \)
79 \( 1 + (-10.8 - 6.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.47 - 12.9i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 3.54T + 89T^{2} \)
97 \( 1 + (-5.79 + 10.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.72447639650479776479368149820, −10.31374818863557113574106195941, −9.196199636993663820579827335405, −8.459152090168403013667930712747, −6.94580883539194600101185811445, −5.81258740936296892035800463306, −5.31179572033432118356553054256, −4.06216764475049771969867876603, −3.59141650316012773198631269690, −1.97752214271376809226723935392, 0.094848767073327582457523207539, 2.40142554402797622557356365555, 3.66637120108861191110414805165, 4.84358257701196628644693489186, 5.78234513092388186338106064205, 6.42231649102219145648517576920, 7.57061976872629107096078244315, 7.84094058816178214445661596818, 8.867164831370269841445477613411, 10.36557526013982276070072731546

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