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Properties

Label 2-810-45.14-c2-0-27
Degree $2$
Conductor $810$
Sign $0.927 + 0.374i$
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 + (−0.173 + 4.99i)5-s + (11.8 − 6.84i)7-s + 2.82·8-s + (6.24 − 3.32i)10-s + (10.6 − 6.15i)11-s + (14.7 + 8.50i)13-s + (−16.7 − 9.67i)14-s + (−2.00 − 3.46i)16-s + 6.89·17-s − 7.24·19-s + (−8.48 − 5.29i)20-s + (−15.0 − 8.70i)22-s + (−17.3 + 30.1i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.0346 + 0.999i)5-s + (1.69 − 0.977i)7-s + 0.353·8-s + (0.624 − 0.332i)10-s + (0.969 − 0.559i)11-s + (1.13 + 0.654i)13-s + (−1.19 − 0.691i)14-s + (−0.125 − 0.216i)16-s + 0.405·17-s − 0.381·19-s + (−0.424 − 0.264i)20-s + (−0.685 − 0.395i)22-s + (−0.756 + 1.31i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.927 + 0.374i$
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: \((2,\ 810,\ (\ :1),\ 0.927 + 0.374i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.054798167\)
\(L(\frac12)\) \(\approx\) \(2.054798167\)
\(L(2)\) 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 \( 1 + (0.173 - 4.99i)T \)
good7 \( 1 + (-11.8 + 6.84i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-10.6 + 6.15i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-14.7 - 8.50i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 6.89T + 289T^{2} \)
19 \( 1 + 7.24T + 361T^{2} \)
23 \( 1 + (17.3 - 30.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (18.3 - 10.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-19.1 + 33.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 21.5iT - 1.36e3T^{2} \)
41 \( 1 + (31.4 + 18.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-5.40 + 3.11i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-20.1 - 34.8i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 38.2T + 2.80e3T^{2} \)
59 \( 1 + (-36.0 - 20.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-7.52 - 13.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-111. - 64.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 104. iT - 5.04e3T^{2} \)
73 \( 1 - 2.11iT - 5.32e3T^{2} \)
79 \( 1 + (-22.0 - 38.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-27.5 - 47.6i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 68.1iT - 7.92e3T^{2} \)
97 \( 1 + (-87.6 + 50.6i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.23150001782819021307245110250, −9.212727070228278294473328050658, −8.250480398121873329902629198779, −7.61123076819140627103038629897, −6.70567751045605391798151332346, −5.57968622060594828872829359910, −4.00927154651713340206162425555, −3.77756894658837106240983324333, −2.03140740518457969547061455779, −1.12439497969789112115104229166, 1.07220884815496910694421035658, 1.99201252381411382649445591788, 4.04741587646894709786066585647, 4.91103450413610078770956029919, 5.62336141031455296229460593563, 6.55835636843376005729585782665, 7.934363852774435211462295239629, 8.431838865446979677784683448950, 8.839621132032531590312986708550, 9.912592827963130609713457412118

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