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Properties

Label 2-720-16.13-c1-0-30
Degree $2$
Conductor $720$
Sign $-0.382 + 0.923i$
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  − 1.41·2-s + 2.00·4-s + (0.707 − 0.707i)5-s − 0.828i·7-s − 2.82·8-s + (−1.00 + 1.00i)10-s + (1.41 − 1.41i)11-s + (−3.41 − 3.41i)13-s + 1.17i·14-s + 4.00·16-s − 2.58·17-s + (−1.82 − 1.82i)19-s + (1.41 − 1.41i)20-s + (−2.00 + 2.00i)22-s − 2.58i·23-s + ⋯
L(s)  = 1  − 1.00·2-s + 1.00·4-s + (0.316 − 0.316i)5-s − 0.313i·7-s − 1.00·8-s + (−0.316 + 0.316i)10-s + (0.426 − 0.426i)11-s + (−0.946 − 0.946i)13-s + 0.313i·14-s + 1.00·16-s − 0.627·17-s + (−0.419 − 0.419i)19-s + (0.316 − 0.316i)20-s + (−0.426 + 0.426i)22-s − 0.539i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.395422 - 0.591791i\)
\(L(\frac12)\) \(\approx\) \(0.395422 - 0.591791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + 0.828iT - 7T^{2} \)
11 \( 1 + (-1.41 + 1.41i)T - 11iT^{2} \)
13 \( 1 + (3.41 + 3.41i)T + 13iT^{2} \)
17 \( 1 + 2.58T + 17T^{2} \)
19 \( 1 + (1.82 + 1.82i)T + 19iT^{2} \)
23 \( 1 + 2.58iT - 23T^{2} \)
29 \( 1 + (-3.41 - 3.41i)T + 29iT^{2} \)
31 \( 1 + 7.65T + 31T^{2} \)
37 \( 1 + (-7.41 + 7.41i)T - 37iT^{2} \)
41 \( 1 + 0.828iT - 41T^{2} \)
43 \( 1 + (7.65 - 7.65i)T - 43iT^{2} \)
47 \( 1 + 7.07T + 47T^{2} \)
53 \( 1 + (-4 + 4i)T - 53iT^{2} \)
59 \( 1 + (-4.58 + 4.58i)T - 59iT^{2} \)
61 \( 1 + (7.48 + 7.48i)T + 61iT^{2} \)
67 \( 1 + (-3.65 - 3.65i)T + 67iT^{2} \)
71 \( 1 + 8iT - 71T^{2} \)
73 \( 1 + 8.82iT - 73T^{2} \)
79 \( 1 + 2.34T + 79T^{2} \)
83 \( 1 + (11.0 + 11.0i)T + 83iT^{2} \)
89 \( 1 + 7.65iT - 89T^{2} \)
97 \( 1 - 9.31T + 97T^{2} \)
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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

−10.07715619797302013719339488391, −9.250761882584273640852069680591, −8.551065094081506901486132462291, −7.65204749285565736797505194865, −6.79334067212668756707761512982, −5.89126667995914317966160372242, −4.75005338284844830900553054216, −3.23304291006816481498824459838, −2.04028655996066984937320819319, −0.48620519971689950062011241633, 1.72625833000394906372956533614, 2.64382980780510428336483881294, 4.16015494190737281487489252668, 5.54204983701888154934496575567, 6.61291326472966388617136616025, 7.13588569585663014114336140843, 8.213220452806908284075620183421, 9.118485178944268604094311051639, 9.726279431357245212204591513425, 10.43702102094457407262062782089

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