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Properties

Label 2-720-45.2-c1-0-20
Degree $2$
Conductor $720$
Sign $0.348 + 0.937i$
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  + (−1.10 − 1.33i)3-s + (0.792 − 2.09i)5-s + (1.05 − 0.283i)7-s + (−0.548 + 2.94i)9-s + (5.44 + 3.14i)11-s + (3.34 + 0.896i)13-s + (−3.66 + 1.25i)15-s + (3.14 − 3.14i)17-s − 1.55i·19-s + (−1.55 − 1.09i)21-s + (−0.258 + 0.965i)23-s + (−3.74 − 3.31i)25-s + (4.53 − 2.53i)27-s + (−1.57 + 2.72i)29-s + (−2.22 − 3.85i)31-s + ⋯
L(s)  = 1  + (−0.639 − 0.769i)3-s + (0.354 − 0.935i)5-s + (0.400 − 0.107i)7-s + (−0.182 + 0.983i)9-s + (1.64 + 0.948i)11-s + (0.928 + 0.248i)13-s + (−0.945 + 0.325i)15-s + (0.763 − 0.763i)17-s − 0.355i·19-s + (−0.338 − 0.239i)21-s + (−0.0539 + 0.201i)23-s + (−0.748 − 0.663i)25-s + (0.872 − 0.487i)27-s + (−0.292 + 0.505i)29-s + (−0.399 − 0.692i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.348 + 0.937i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.348 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25455 - 0.871609i\)
\(L(\frac12)\) \(\approx\) \(1.25455 - 0.871609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.10 + 1.33i)T \)
5 \( 1 + (-0.792 + 2.09i)T \)
good7 \( 1 + (-1.05 + 0.283i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-5.44 - 3.14i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.34 - 0.896i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-3.14 + 3.14i)T - 17iT^{2} \)
19 \( 1 + 1.55iT - 19T^{2} \)
23 \( 1 + (0.258 - 0.965i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.57 - 2.72i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.22 + 3.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 + (3.39 - 1.96i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.896 + 3.34i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.32 + 8.69i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.61 - 6.61i)T + 53iT^{2} \)
59 \( 1 + (-5.90 - 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.72 + 4.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.978 - 3.65i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.635iT - 71T^{2} \)
73 \( 1 + (-2.89 + 2.89i)T - 73iT^{2} \)
79 \( 1 + (2.12 + 1.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.531 + 0.142i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 2.36T + 89T^{2} \)
97 \( 1 + (-10.7 + 2.89i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.23084538520184278275259126385, −9.272493941762794124561985412105, −8.601245977567894423327363144588, −7.48050480391426009165806767622, −6.74109468099192778747557768988, −5.77212304504437341149225672687, −4.92793852246534286196181039502, −3.91931703893039862341005045040, −1.90167968012703380951272818188, −1.08388334578722462950625355891, 1.40001341998620602543489607870, 3.38560982027414113342567854533, 3.85920943645874940210307653602, 5.35570501066652036097349244571, 6.18054489515975045897298305336, 6.65442490233123946395454229886, 8.153022016405837514825485963616, 8.975361609142908143903474006617, 9.906573231578370639008988589306, 10.60595857366989085188840666337

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