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Properties

Label 2-720-45.23-c1-0-10
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: $\chi_{720} (113, \cdot )$
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

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

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