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Properties

Label 2-720-15.2-c1-0-8
Degree $2$
Conductor $720$
Sign $0.374 + 0.927i$
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  + (1.58 − 1.58i)5-s + (−1.23 − 1.23i)7-s + 1.74i·11-s + (0.236 − 0.236i)13-s + (4.57 − 4.57i)17-s − 6.47i·19-s + (2.82 + 2.82i)23-s − 5.00i·25-s + 0.333·29-s − 10.4·31-s − 3.90·35-s + (−2.23 − 2.23i)37-s + 7.07i·41-s + (6.47 − 6.47i)43-s + (4.57 − 4.57i)47-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s + (−0.467 − 0.467i)7-s + 0.527i·11-s + (0.0654 − 0.0654i)13-s + (1.10 − 1.10i)17-s − 1.48i·19-s + (0.589 + 0.589i)23-s − 1.00i·25-s + 0.0619·29-s − 1.88·31-s − 0.660·35-s + (−0.367 − 0.367i)37-s + 1.10i·41-s + (0.986 − 0.986i)43-s + (0.667 − 0.667i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.374 + 0.927i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30426 - 0.880041i\)
\(L(\frac12)\) \(\approx\) \(1.30426 - 0.880041i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 + 1.58i)T \)
good7 \( 1 + (1.23 + 1.23i)T + 7iT^{2} \)
11 \( 1 - 1.74iT - 11T^{2} \)
13 \( 1 + (-0.236 + 0.236i)T - 13iT^{2} \)
17 \( 1 + (-4.57 + 4.57i)T - 17iT^{2} \)
19 \( 1 + 6.47iT - 19T^{2} \)
23 \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \)
29 \( 1 - 0.333T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + (2.23 + 2.23i)T + 37iT^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + (-6.47 + 6.47i)T - 43iT^{2} \)
47 \( 1 + (-4.57 + 4.57i)T - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 - 7.40T + 59T^{2} \)
61 \( 1 - 1.52T + 61T^{2} \)
67 \( 1 + (-10.4 - 10.4i)T + 67iT^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 + (9.47 - 9.47i)T - 73iT^{2} \)
79 \( 1 + 5.52iT - 79T^{2} \)
83 \( 1 + (-7.40 - 7.40i)T + 83iT^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (-1 - i)T + 97iT^{2} \)
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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

−10.04899156582422335034665934484, −9.419403904158032992757345140676, −8.794158923496675010782406716697, −7.44127741773618261757157207090, −6.89679500378652410586631864983, −5.55696472464395569016512720113, −4.98152845454827976155942355057, −3.69946012203641923296787048163, −2.40050616504967453968510800726, −0.865984570129508305178368537061, 1.67039373047832472150234848951, 2.98438082697870441577400508007, 3.85448818350282352965191463662, 5.64723994970454584999424086523, 5.89406892537978868026529463509, 6.98944343696594347636147043269, 7.995084968669466924988123527333, 8.955561301771092782766026180746, 9.799775084837208694009631348134, 10.52815515269483667544521492826

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