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Properties

Label 2-720-15.14-c2-0-9
Degree $2$
Conductor $720$
Sign $0.998 - 0.0519i$
Analytic cond. $19.6185$
Root an. cond. $4.42928$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.09 − 3.92i)5-s + 0.822i·7-s + 5.93i·11-s + 11.9i·13-s − 14.0·17-s + 21.5·19-s + 26.9·23-s + (−5.84 + 24.3i)25-s − 26.9i·29-s + 27.3·31-s + (3.23 − 2.54i)35-s + 10.8i·37-s − 47.5i·41-s − 19.0i·43-s + 89.1·47-s + ⋯
L(s)  = 1  + (−0.618 − 0.785i)5-s + 0.117i·7-s + 0.539i·11-s + 0.917i·13-s − 0.824·17-s + 1.13·19-s + 1.17·23-s + (−0.233 + 0.972i)25-s − 0.929i·29-s + 0.881·31-s + (0.0923 − 0.0727i)35-s + 0.294i·37-s − 1.15i·41-s − 0.443i·43-s + 1.89·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0519i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.998 - 0.0519i$
Analytic conductor: \(19.6185\)
Root analytic conductor: \(4.42928\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1),\ 0.998 - 0.0519i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.569733138\)
\(L(\frac12)\) \(\approx\) \(1.569733138\)
\(L(2)\) 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 \( 1 + (3.09 + 3.92i)T \)
good7 \( 1 - 0.822iT - 49T^{2} \)
11 \( 1 - 5.93iT - 121T^{2} \)
13 \( 1 - 11.9iT - 169T^{2} \)
17 \( 1 + 14.0T + 289T^{2} \)
19 \( 1 - 21.5T + 361T^{2} \)
23 \( 1 - 26.9T + 529T^{2} \)
29 \( 1 + 26.9iT - 841T^{2} \)
31 \( 1 - 27.3T + 961T^{2} \)
37 \( 1 - 10.8iT - 1.36e3T^{2} \)
41 \( 1 + 47.5iT - 1.68e3T^{2} \)
43 \( 1 + 19.0iT - 1.84e3T^{2} \)
47 \( 1 - 89.1T + 2.20e3T^{2} \)
53 \( 1 - 15.3T + 2.80e3T^{2} \)
59 \( 1 - 15.6iT - 3.48e3T^{2} \)
61 \( 1 - 17.6T + 3.72e3T^{2} \)
67 \( 1 - 116. iT - 4.48e3T^{2} \)
71 \( 1 + 84.4iT - 5.04e3T^{2} \)
73 \( 1 - 79.0iT - 5.32e3T^{2} \)
79 \( 1 - 94.9T + 6.24e3T^{2} \)
83 \( 1 + 74.3T + 6.88e3T^{2} \)
89 \( 1 - 90.7iT - 7.92e3T^{2} \)
97 \( 1 + 141. iT - 9.40e3T^{2} \)
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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

−10.13299449517919735763896037668, −9.148022575487812149388333250825, −8.704908062989409847858659370200, −7.53106829327361347363541970549, −6.90607273951288477838107447966, −5.59362378847504304698677553401, −4.65437814534231219369086321913, −3.89092625803490187464202616929, −2.39104390845829924636407095541, −0.917947560587468141852848191221, 0.78855771578154660031043502641, 2.72830620126721884604052081189, 3.45716111607789966458649552428, 4.69014477437059284876380755234, 5.78272888424273278275920383573, 6.82084200390352461621662704058, 7.53674214462236273691808541882, 8.396885837640545987207775679280, 9.331289755336329138038926427928, 10.43763479122412624939899521619

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