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Properties

Label 2-720-60.47-c0-0-0
Degree $2$
Conductor $720$
Sign $0.927 - 0.374i$
Analytic cond. $0.359326$
Root an. cond. $0.599438$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)5-s + (1 − i)13-s + (−1.41 + 1.41i)17-s + 1.00i·25-s + 1.41·29-s + (−1 − i)37-s − 1.41i·41-s + i·49-s + (−1.41 − 1.41i)53-s + 1.41·65-s + (−1 + i)73-s − 2.00·85-s − 1.41·89-s + (−1 − i)97-s − 1.41i·101-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)5-s + (1 − i)13-s + (−1.41 + 1.41i)17-s + 1.00i·25-s + 1.41·29-s + (−1 − i)37-s − 1.41i·41-s + i·49-s + (−1.41 − 1.41i)53-s + 1.41·65-s + (−1 + i)73-s − 2.00·85-s − 1.41·89-s + (−1 − i)97-s − 1.41i·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :0),\ 0.927 - 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066328661\)
\(L(\frac12)\) \(\approx\) \(1.066328661\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
17 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−10.71583165439973006758104148918, −10.01406325985155680674132560920, −8.861739567397791545992470839305, −8.248060925027745167681633201336, −6.99653012123343644775849492610, −6.25047255963373333986948750064, −5.50816170288864579214599052022, −4.11886355285127020876414119677, −3.03630434389988737926753874782, −1.79875325280363937898677624188, 1.48032047084430832667545811613, 2.79133556326799713206313610323, 4.36013492501150072990291401627, 5.00197457637456366638609138068, 6.28110856756256488084407364861, 6.81254689025469872329129787550, 8.222892346881707750128076094486, 8.953670645370411018823123301450, 9.522693860509059120906141963969, 10.53305327494656311680483445437

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