Properties

Label 2-720-80.69-c1-0-3
Degree $2$
Conductor $720$
Sign $-0.998 - 0.0618i$
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  + (−0.345 + 1.37i)2-s + (−1.76 − 0.946i)4-s + (0.561 − 2.16i)5-s − 4.51·7-s + (1.90 − 2.08i)8-s + (2.77 + 1.51i)10-s + (3.44 + 3.44i)11-s + (0.113 + 0.113i)13-s + (1.55 − 6.19i)14-s + (2.20 + 3.33i)16-s + 5.03i·17-s + (−0.992 + 0.992i)19-s + (−3.03 + 3.28i)20-s + (−5.90 + 3.53i)22-s − 8.00·23-s + ⋯
L(s)  = 1  + (−0.244 + 0.969i)2-s + (−0.880 − 0.473i)4-s + (0.251 − 0.967i)5-s − 1.70·7-s + (0.673 − 0.738i)8-s + (0.877 + 0.479i)10-s + (1.03 + 1.03i)11-s + (0.0315 + 0.0315i)13-s + (0.416 − 1.65i)14-s + (0.551 + 0.833i)16-s + 1.22i·17-s + (−0.227 + 0.227i)19-s + (−0.679 + 0.733i)20-s + (−1.25 + 0.752i)22-s − 1.66·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.998 - 0.0618i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.998 - 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0132169 + 0.426765i\)
\(L(\frac12)\) \(\approx\) \(0.0132169 + 0.426765i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.345 - 1.37i)T \)
3 \( 1 \)
5 \( 1 + (-0.561 + 2.16i)T \)
good7 \( 1 + 4.51T + 7T^{2} \)
11 \( 1 + (-3.44 - 3.44i)T + 11iT^{2} \)
13 \( 1 + (-0.113 - 0.113i)T + 13iT^{2} \)
17 \( 1 - 5.03iT - 17T^{2} \)
19 \( 1 + (0.992 - 0.992i)T - 19iT^{2} \)
23 \( 1 + 8.00T + 23T^{2} \)
29 \( 1 + (-1.01 + 1.01i)T - 29iT^{2} \)
31 \( 1 + 6.42T + 31T^{2} \)
37 \( 1 + (1.63 - 1.63i)T - 37iT^{2} \)
41 \( 1 - 3.35iT - 41T^{2} \)
43 \( 1 + (5.68 - 5.68i)T - 43iT^{2} \)
47 \( 1 - 9.10iT - 47T^{2} \)
53 \( 1 + (-3.27 + 3.27i)T - 53iT^{2} \)
59 \( 1 + (-5.30 - 5.30i)T + 59iT^{2} \)
61 \( 1 + (5.87 - 5.87i)T - 61iT^{2} \)
67 \( 1 + (-1.87 - 1.87i)T + 67iT^{2} \)
71 \( 1 + 0.635iT - 71T^{2} \)
73 \( 1 + 6.14T + 73T^{2} \)
79 \( 1 - 1.76T + 79T^{2} \)
83 \( 1 + (6.39 + 6.39i)T + 83iT^{2} \)
89 \( 1 - 0.579iT - 89T^{2} \)
97 \( 1 + 15.2iT - 97T^{2} \)
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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

−10.20272561181714120871864960602, −9.811032105007700241166163118872, −9.115879510820936889111876596905, −8.333059227788376592251849198802, −7.25463540527916886549770909299, −6.25895675412742427245783894618, −5.91168657745499733714441451047, −4.45213891875160410407251474851, −3.77069054803586022581321469481, −1.62013692475495836549878740465, 0.23863433213088286075219553251, 2.24636418576468631955158661812, 3.36347343360455087967066752662, 3.77337592845243245063266979624, 5.57720983054961636587770016110, 6.49879852694152929597096135363, 7.27930887873626706762600626144, 8.653616723312224564453688625138, 9.403430327271716334368101740632, 9.995213953526529410737664417013

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