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Properties

Label 2-720-15.8-c1-0-3
Degree $2$
Conductor $720$
Sign $0.749 - 0.662i$
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  + (−0.707 − 2.12i)5-s + (−2 + 2i)7-s + 2.82i·11-s + (3 + 3i)13-s + (1.41 + 1.41i)17-s + 4i·19-s + (5.65 − 5.65i)23-s + (−3.99 + 3i)25-s + 9.89·29-s + 8·31-s + (5.65 + 2.82i)35-s + (−3 + 3i)37-s + 1.41i·41-s + (−8.48 − 8.48i)47-s − i·49-s + ⋯
L(s)  = 1  + (−0.316 − 0.948i)5-s + (−0.755 + 0.755i)7-s + 0.852i·11-s + (0.832 + 0.832i)13-s + (0.342 + 0.342i)17-s + 0.917i·19-s + (1.17 − 1.17i)23-s + (−0.799 + 0.600i)25-s + 1.83·29-s + 1.43·31-s + (0.956 + 0.478i)35-s + (−0.493 + 0.493i)37-s + 0.220i·41-s + (−1.23 − 1.23i)47-s − 0.142i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.749 - 0.662i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20666 + 0.456597i\)
\(L(\frac12)\) \(\approx\) \(1.20666 + 0.456597i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 2.12i)T \)
good7 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-5.65 + 5.65i)T - 23iT^{2} \)
29 \( 1 - 9.89T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (8.48 + 8.48i)T + 47iT^{2} \)
53 \( 1 + (7.07 - 7.07i)T - 53iT^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + (8 - 8i)T - 67iT^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (-7 - 7i)T + 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 + 1.41T + 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−10.30620996519576443828090402970, −9.630367804600779296830162536194, −8.645161673239013648032660232310, −8.280394592745292990785662355693, −6.84015897009788873126815544347, −6.15554584833467913489118584467, −4.97099835435441550210500193027, −4.16675319724783795628503102288, −2.86545373662246088416561761598, −1.34648498631240175762356989187, 0.77266078706127326832727693774, 3.08606963579875404429779005416, 3.33096957499814993653475337940, 4.80131505047026589461807564105, 6.12559248645424503146381628040, 6.73399695271106497869699011571, 7.64124629861455425802656665615, 8.500778741625009362052580362275, 9.620507531795024041509552879623, 10.43384567398867937493607818605

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