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Properties

Label 2-720-60.59-c5-0-52
Degree $2$
Conductor $720$
Sign $-0.379 + 0.925i$
Analytic cond. $115.476$
Root an. cond. $10.7459$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (25.8 − 49.5i)5-s + 154.·7-s + 104.·11-s − 727. i·13-s + 484.·17-s − 877. i·19-s − 1.02e3i·23-s + (−1.78e3 − 2.56e3i)25-s − 6.37e3i·29-s + 7.63e3i·31-s + (4.00e3 − 7.67e3i)35-s − 5.54e3i·37-s + 4.74e3i·41-s − 4.32e3·43-s − 1.62e4i·47-s + ⋯
L(s)  = 1  + (0.463 − 0.886i)5-s + 1.19·7-s + 0.259·11-s − 1.19i·13-s + 0.406·17-s − 0.557i·19-s − 0.404i·23-s + (−0.571 − 0.820i)25-s − 1.40i·29-s + 1.42i·31-s + (0.553 − 1.05i)35-s − 0.666i·37-s + 0.440i·41-s − 0.357·43-s − 1.07i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.379 + 0.925i$
Analytic conductor: \(115.476\)
Root analytic conductor: \(10.7459\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :5/2),\ -0.379 + 0.925i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.659974176\)
\(L(\frac12)\) \(\approx\) \(2.659974176\)
\(L(\frac{7}{2})\) 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 \( 1 + (-25.8 + 49.5i)T \)
good7 \( 1 - 154.T + 1.68e4T^{2} \)
11 \( 1 - 104.T + 1.61e5T^{2} \)
13 \( 1 + 727. iT - 3.71e5T^{2} \)
17 \( 1 - 484.T + 1.41e6T^{2} \)
19 \( 1 + 877. iT - 2.47e6T^{2} \)
23 \( 1 + 1.02e3iT - 6.43e6T^{2} \)
29 \( 1 + 6.37e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.63e3iT - 2.86e7T^{2} \)
37 \( 1 + 5.54e3iT - 6.93e7T^{2} \)
41 \( 1 - 4.74e3iT - 1.15e8T^{2} \)
43 \( 1 + 4.32e3T + 1.47e8T^{2} \)
47 \( 1 + 1.62e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.18e3T + 4.18e8T^{2} \)
59 \( 1 - 1.34e4T + 7.14e8T^{2} \)
61 \( 1 + 1.01e4T + 8.44e8T^{2} \)
67 \( 1 - 4.07e4T + 1.35e9T^{2} \)
71 \( 1 - 2.99e4T + 1.80e9T^{2} \)
73 \( 1 - 4.25e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.70e4iT - 3.07e9T^{2} \)
83 \( 1 - 9.78e4iT - 3.93e9T^{2} \)
89 \( 1 - 5.41e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.24e5iT - 8.58e9T^{2} \)
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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

−9.349272420198239954082359045232, −8.331189040535399099660548399197, −8.004805340440838348663106510497, −6.72148792682036245532283884615, −5.50362294790523100492203175478, −5.03980527721180105990086575346, −3.99429567378212386378450842039, −2.54650000098733021263298904908, −1.43261307306698712357843783044, −0.54582766717048438309778450237, 1.38156910854272029860715354340, 2.10530853271871728982738225407, 3.41013045853274778401782637586, 4.46965437835555811962045591684, 5.49697801101376124212857286585, 6.44101510659407872851863056435, 7.30061967343110205618636261178, 8.104963006038570238336341154875, 9.153204940425013838414914294179, 9.905190403380359303119077878318

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