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Properties

Label 2-720-9.2-c2-0-29
Degree $2$
Conductor $720$
Sign $-0.142 + 0.989i$
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  + (−2.62 − 1.45i)3-s + (1.93 − 1.11i)5-s + (−3.63 + 6.28i)7-s + (4.74 + 7.65i)9-s + (−6.50 − 3.75i)11-s + (−1.46 − 2.54i)13-s + (−6.70 + 0.104i)15-s + 1.90i·17-s + 7.38·19-s + (18.6 − 11.1i)21-s + (30.6 − 17.6i)23-s + (2.5 − 4.33i)25-s + (−1.26 − 26.9i)27-s + (−14.2 − 8.19i)29-s + (13.3 + 23.0i)31-s + ⋯
L(s)  = 1  + (−0.873 − 0.486i)3-s + (0.387 − 0.223i)5-s + (−0.518 + 0.898i)7-s + (0.526 + 0.850i)9-s + (−0.591 − 0.341i)11-s + (−0.112 − 0.195i)13-s + (−0.447 + 0.00697i)15-s + 0.111i·17-s + 0.388·19-s + (0.890 − 0.532i)21-s + (1.33 − 0.769i)23-s + (0.100 − 0.173i)25-s + (−0.0467 − 0.998i)27-s + (−0.489 − 0.282i)29-s + (0.429 + 0.744i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.142 + 0.989i$
Analytic conductor: \(19.6185\)
Root analytic conductor: \(4.42928\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1),\ -0.142 + 0.989i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9198349934\)
\(L(\frac12)\) \(\approx\) \(0.9198349934\)
\(L(2)\) 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 \( 1 + (2.62 + 1.45i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
good7 \( 1 + (3.63 - 6.28i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (6.50 + 3.75i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (1.46 + 2.54i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 1.90iT - 289T^{2} \)
19 \( 1 - 7.38T + 361T^{2} \)
23 \( 1 + (-30.6 + 17.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (14.2 + 8.19i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-13.3 - 23.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 44.6T + 1.36e3T^{2} \)
41 \( 1 + (-14.8 + 8.58i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-20.5 + 35.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (36.4 + 21.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 100. iT - 2.80e3T^{2} \)
59 \( 1 + (-4.12 + 2.37i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-38.4 + 66.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (41.9 + 72.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 23.1iT - 5.04e3T^{2} \)
73 \( 1 + 103.T + 5.32e3T^{2} \)
79 \( 1 + (-40.2 + 69.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-17.1 - 9.87i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 29.0iT - 7.92e3T^{2} \)
97 \( 1 + (-47.7 + 82.7i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.12499787556257293347270390093, −9.098007570980889162173156123178, −8.287158874962475488663976103431, −7.15809041327613988448443667521, −6.34317914752982108307841656224, −5.48198444825836918040260476277, −4.90266161243580826805516317126, −3.15092600423931385342117329524, −1.97293763870801400466326224046, −0.41515032830256198593395577136, 1.12686464899657264193512949847, 2.96388300668206577856710028263, 4.08832370770431772325609187718, 5.06432038030518858758661209179, 5.93033413118712515526857339202, 6.95414654651313372849313411923, 7.49146524616922513682799143419, 9.072549405849541591500733218121, 9.776374626885309167292012121750, 10.43730741272912532018564344240

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