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Properties

Label 2-720-20.3-c3-0-23
Degree $2$
Conductor $720$
Sign $0.930 + 0.365i$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 11i)5-s + (55 + 55i)13-s + (5 − 5i)17-s + (−117 + 44i)25-s + 284i·29-s + (305 − 305i)37-s + 472·41-s − 343i·49-s + (−545 − 545i)53-s + 468·61-s + (495 − 715i)65-s + (845 + 845i)73-s + (−65 − 45i)85-s − 176i·89-s + (1.20e3 − 1.20e3i)97-s + ⋯
L(s)  = 1  + (−0.178 − 0.983i)5-s + (1.17 + 1.17i)13-s + (0.0713 − 0.0713i)17-s + (−0.936 + 0.351i)25-s + 1.81i·29-s + (1.35 − 1.35i)37-s + 1.79·41-s − i·49-s + (−1.41 − 1.41i)53-s + 0.982·61-s + (0.944 − 1.36i)65-s + (1.35 + 1.35i)73-s + (−0.0829 − 0.0574i)85-s − 0.209i·89-s + (1.26 − 1.26i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.930 + 0.365i$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ 0.930 + 0.365i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.019336324\)
\(L(\frac12)\) \(\approx\) \(2.019336324\)
\(L(\frac{5}{2})\) 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 \( 1 + (2 + 11i)T \)
good7 \( 1 + 343iT^{2} \)
11 \( 1 - 1.33e3T^{2} \)
13 \( 1 + (-55 - 55i)T + 2.19e3iT^{2} \)
17 \( 1 + (-5 + 5i)T - 4.91e3iT^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 - 1.21e4iT^{2} \)
29 \( 1 - 284iT - 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 + (-305 + 305i)T - 5.06e4iT^{2} \)
41 \( 1 - 472T + 6.89e4T^{2} \)
43 \( 1 - 7.95e4iT^{2} \)
47 \( 1 + 1.03e5iT^{2} \)
53 \( 1 + (545 + 545i)T + 1.48e5iT^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 468T + 2.26e5T^{2} \)
67 \( 1 + 3.00e5iT^{2} \)
71 \( 1 - 3.57e5T^{2} \)
73 \( 1 + (-845 - 845i)T + 3.89e5iT^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 - 5.71e5iT^{2} \)
89 \( 1 + 176iT - 7.04e5T^{2} \)
97 \( 1 + (-1.20e3 + 1.20e3i)T - 9.12e5iT^{2} \)
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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

−9.752539448161338722711781985564, −9.021038256292121561963168621617, −8.434774163510553721743460744860, −7.40367563835019125982729736841, −6.38090626175692947072100751376, −5.42365980623551616013159577834, −4.41691765596952084384104064892, −3.58793458086556101685615658429, −1.93469832411392515147513232177, −0.827088644139587172965063481575, 0.844783072068396638577751644417, 2.50309463173715131180934963601, 3.40502973650737013732085776326, 4.43262118835111899270036255911, 5.94069092019349706860256077472, 6.30309606982893415500692894033, 7.70037538277589803902575063093, 8.035207237350253545276539554942, 9.327099308316052791764916078768, 10.17090362423516289352292614470

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