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Properties

Label 2-720-16.5-c1-0-12
Degree $2$
Conductor $720$
Sign $-0.594 - 0.804i$
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  + (0.0861 + 1.41i)2-s + (−1.98 + 0.243i)4-s + (0.707 + 0.707i)5-s − 2.76i·7-s + (−0.514 − 2.78i)8-s + (−0.937 + 1.05i)10-s + (3.51 + 3.51i)11-s + (−4.55 + 4.55i)13-s + (3.90 − 0.238i)14-s + (3.88 − 0.965i)16-s + 5.00·17-s + (−0.812 + 0.812i)19-s + (−1.57 − 1.23i)20-s + (−4.65 + 5.25i)22-s + 7.48i·23-s + ⋯
L(s)  = 1  + (0.0609 + 0.998i)2-s + (−0.992 + 0.121i)4-s + (0.316 + 0.316i)5-s − 1.04i·7-s + (−0.181 − 0.983i)8-s + (−0.296 + 0.334i)10-s + (1.05 + 1.05i)11-s + (−1.26 + 1.26i)13-s + (1.04 − 0.0636i)14-s + (0.970 − 0.241i)16-s + 1.21·17-s + (−0.186 + 0.186i)19-s + (−0.352 − 0.275i)20-s + (−0.991 + 1.12i)22-s + 1.56i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.594 - 0.804i$
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: \((2,\ 720,\ (\ :1/2),\ -0.594 - 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620792 + 1.23092i\)
\(L(\frac12)\) \(\approx\) \(0.620792 + 1.23092i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0861 - 1.41i)T \)
3 \( 1 \)
5 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + 2.76iT - 7T^{2} \)
11 \( 1 + (-3.51 - 3.51i)T + 11iT^{2} \)
13 \( 1 + (4.55 - 4.55i)T - 13iT^{2} \)
17 \( 1 - 5.00T + 17T^{2} \)
19 \( 1 + (0.812 - 0.812i)T - 19iT^{2} \)
23 \( 1 - 7.48iT - 23T^{2} \)
29 \( 1 + (6.03 - 6.03i)T - 29iT^{2} \)
31 \( 1 - 7.58T + 31T^{2} \)
37 \( 1 + (-1.08 - 1.08i)T + 37iT^{2} \)
41 \( 1 + 3.15iT - 41T^{2} \)
43 \( 1 + (3.10 + 3.10i)T + 43iT^{2} \)
47 \( 1 - 2.76T + 47T^{2} \)
53 \( 1 + (6.41 + 6.41i)T + 53iT^{2} \)
59 \( 1 + (-5.13 - 5.13i)T + 59iT^{2} \)
61 \( 1 + (2.49 - 2.49i)T - 61iT^{2} \)
67 \( 1 + (3.14 - 3.14i)T - 67iT^{2} \)
71 \( 1 - 3.50iT - 71T^{2} \)
73 \( 1 + 14.6iT - 73T^{2} \)
79 \( 1 - 8.95T + 79T^{2} \)
83 \( 1 + (-2.86 + 2.86i)T - 83iT^{2} \)
89 \( 1 - 7.23iT - 89T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
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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

−10.31737334878277528751319709963, −9.676090265894956020494769227238, −9.167258351895788987273663105169, −7.69866818219330274303370229430, −7.17487635215323691355101056129, −6.60349814887235927157496591268, −5.35664112151914414018351894189, −4.40801918185281712718431661721, −3.57810799088159817710183353312, −1.56079164239271494737924116737, 0.76910358289905036605869069879, 2.37999238958374438846783609540, 3.20074416797304444737617085817, 4.54576580927102264986582493240, 5.51896686406791183687577329237, 6.17437361263606799953100115398, 7.936247109177535216952129980399, 8.548419628089878195898517134337, 9.476131820952346335892897942557, 10.01027550693425677761602647982

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