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Properties

Label 2-810-45.14-c2-0-25
Degree $2$
Conductor $810$
Sign $0.700 + 0.713i$
Analytic cond. $22.0709$
Root an. cond. $4.69796$
Motivic weight $2$
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 + 1.22i)2-s + (−0.999 + 1.73i)4-s + (−4.98 + 0.387i)5-s + (−5.04 + 2.91i)7-s − 2.82·8-s + (−4 − 5.83i)10-s + (−14.2 + 8.24i)11-s + (−7.14 − 4.12i)14-s + (−2.00 − 3.46i)16-s + 11.3·17-s + 12·19-s + (4.31 − 9.02i)20-s + (−20.1 − 11.6i)22-s + (12.0 − 20.8i)23-s + (24.6 − 3.86i)25-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.996 + 0.0775i)5-s + (−0.721 + 0.416i)7-s − 0.353·8-s + (−0.400 − 0.583i)10-s + (−1.29 + 0.749i)11-s + (−0.510 − 0.294i)14-s + (−0.125 − 0.216i)16-s + 0.665·17-s + 0.631·19-s + (0.215 − 0.451i)20-s + (−0.918 − 0.530i)22-s + (0.522 − 0.905i)23-s + (0.987 − 0.154i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.700 + 0.713i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ 0.700 + 0.713i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6500535319\)
\(L(\frac12)\) \(\approx\) \(0.6500535319\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 \)
5 \( 1 + (4.98 - 0.387i)T \)
good7 \( 1 + (5.04 - 2.91i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (14.2 - 8.24i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (84.5 + 146. i)T^{2} \)
17 \( 1 - 11.3T + 289T^{2} \)
19 \( 1 - 12T + 361T^{2} \)
23 \( 1 + (-12.0 + 20.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-16 + 27.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 23.3iT - 1.36e3T^{2} \)
41 \( 1 + (49.9 + 28.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-35.3 + 20.4i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (17.6 + 30.6i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 67.8T + 2.80e3T^{2} \)
59 \( 1 + (-14.2 - 8.24i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-8 - 13.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (5.04 + 2.91i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 116. iT - 5.32e3T^{2} \)
79 \( 1 + (-36 - 62.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (21.9 + 37.9i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 65.9iT - 7.92e3T^{2} \)
97 \( 1 + (-141. + 81.6i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.913245300391675637781695244727, −8.937217350543574779313266781846, −7.971986156774180970514669912957, −7.42923210865706912968348864766, −6.54142865868398262272350418344, −5.44247324089570654621718918584, −4.65360033322925092869548825298, −3.50868738563641401650157663758, −2.60217272322305084075786901877, −0.23397663589609460769690268594, 1.04097387343853413812455076275, 3.05933313280194389489404774109, 3.36431801859981159797849388747, 4.69573674169485759487427966573, 5.50211470880852479765392109869, 6.70057881716731183364851710686, 7.70223623449658372040521438564, 8.376856332044046493147665282526, 9.542984525919805124597692931814, 10.28162667693139687339479088364

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