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Properties

Label 2-720-16.5-c1-0-5
Degree $2$
Conductor $720$
Sign $0.332 - 0.943i$
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  + (0.257 − 1.39i)2-s + (−1.86 − 0.715i)4-s + (0.707 + 0.707i)5-s + 2.89i·7-s + (−1.47 + 2.41i)8-s + (1.16 − 0.801i)10-s + (−1.84 − 1.84i)11-s + (−3.08 + 3.08i)13-s + (4.02 + 0.744i)14-s + (2.97 + 2.67i)16-s − 7.29·17-s + (−1.23 + 1.23i)19-s + (−0.814 − 1.82i)20-s + (−3.03 + 2.09i)22-s + 4.60i·23-s + ⋯
L(s)  = 1  + (0.181 − 0.983i)2-s + (−0.933 − 0.357i)4-s + (0.316 + 0.316i)5-s + 1.09i·7-s + (−0.521 + 0.853i)8-s + (0.368 − 0.253i)10-s + (−0.556 − 0.556i)11-s + (−0.854 + 0.854i)13-s + (1.07 + 0.198i)14-s + (0.744 + 0.667i)16-s − 1.77·17-s + (−0.283 + 0.283i)19-s + (−0.182 − 0.408i)20-s + (−0.648 + 0.445i)22-s + 0.960i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.332 - 0.943i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.332 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.611461 + 0.432890i\)
\(L(\frac12)\) \(\approx\) \(0.611461 + 0.432890i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.257 + 1.39i)T \)
3 \( 1 \)
5 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 - 2.89iT - 7T^{2} \)
11 \( 1 + (1.84 + 1.84i)T + 11iT^{2} \)
13 \( 1 + (3.08 - 3.08i)T - 13iT^{2} \)
17 \( 1 + 7.29T + 17T^{2} \)
19 \( 1 + (1.23 - 1.23i)T - 19iT^{2} \)
23 \( 1 - 4.60iT - 23T^{2} \)
29 \( 1 + (4.24 - 4.24i)T - 29iT^{2} \)
31 \( 1 - 2.06T + 31T^{2} \)
37 \( 1 + (1.17 + 1.17i)T + 37iT^{2} \)
41 \( 1 + 4.61iT - 41T^{2} \)
43 \( 1 + (-3.03 - 3.03i)T + 43iT^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + (2.73 + 2.73i)T + 53iT^{2} \)
59 \( 1 + (3.11 + 3.11i)T + 59iT^{2} \)
61 \( 1 + (-2.34 + 2.34i)T - 61iT^{2} \)
67 \( 1 + (-8.24 + 8.24i)T - 67iT^{2} \)
71 \( 1 - 3.25iT - 71T^{2} \)
73 \( 1 - 12.6iT - 73T^{2} \)
79 \( 1 + 0.113T + 79T^{2} \)
83 \( 1 + (9.76 - 9.76i)T - 83iT^{2} \)
89 \( 1 - 3.74iT - 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−10.84615422438651211530721917490, −9.661329942179310225671845319979, −9.118666432959838495680045625587, −8.368162367980291889313202859815, −6.99678238317171408911762280358, −5.85257241332975815744081316931, −5.11732473938122436588742711723, −3.98093358713209791533284417513, −2.65222671798630532019871448267, −2.00501097347086357464200022044, 0.34422380319236283634086261087, 2.54810012108075979262261428634, 4.21826518173528086448071819414, 4.70683734956335800717659609775, 5.81893523630879333553382591367, 6.86249355676379910291864337949, 7.46853826095626279887660839406, 8.356067072982440702482805963891, 9.268324804928244793067466811029, 10.14862283839926565260830817603

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