Properties

Degree $2$
Conductor $720$
Sign $-0.894 + 0.447i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + i)5-s + 4i·7-s − 4·11-s − 4i·13-s − 6i·17-s − 4·19-s − 4i·23-s + (3 − 4i)25-s − 4·29-s + (−4 − 8i)35-s + 4i·37-s − 8·41-s + 12i·47-s − 9·49-s + 2i·53-s + ⋯
L(s)  = 1  + (−0.894 + 0.447i)5-s + 1.51i·7-s − 1.20·11-s − 1.10i·13-s − 1.45i·17-s − 0.917·19-s − 0.834i·23-s + (0.600 − 0.800i)25-s − 0.742·29-s + (−0.676 − 1.35i)35-s + 0.657i·37-s − 1.24·41-s + 1.75i·47-s − 1.28·49-s + 0.274i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.894 + 0.447i$
Motivic weight: \(1\)
Character: $\chi_{720} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (2 - i)T \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−10.15868904826550110521612182023, −9.058722085716194789880205596641, −8.226901402151902866738141356653, −7.64062808339153370273001563858, −6.48770660992820850167625129137, −5.46364590272685055679858231450, −4.68923333742045961117131458509, −3.07811712083300008315310753009, −2.52872083637152739966316508117, 0, 1.73120086306290826165577023814, 3.63602814170657235871932812636, 4.17257174932616932640189191169, 5.20380729829409182010252153007, 6.57419690303017447639645291802, 7.42891881658125989292916077155, 8.064326474338784466073449581357, 8.926648884687185212925284558356, 10.14648392519205838162628278275

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