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Properties

Label 2-810-45.29-c2-0-17
Degree $2$
Conductor $810$
Sign $-0.471 - 0.881i$
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 + (3.08 + 3.93i)5-s + (9.52 + 5.5i)7-s + 2.82·8-s + (−7 + 0.999i)10-s + (6.12 + 3.53i)11-s + (−12.9 + 7.5i)13-s + (−13.4 + 7.77i)14-s + (−2.00 + 3.46i)16-s + 22.6·17-s − 3·19-s + (3.72 − 9.28i)20-s + (−8.66 + 4.99i)22-s + (9.19 + 15.9i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.617 + 0.786i)5-s + (1.36 + 0.785i)7-s + 0.353·8-s + (−0.700 + 0.0999i)10-s + (0.556 + 0.321i)11-s + (−0.999 + 0.576i)13-s + (−0.962 + 0.555i)14-s + (−0.125 + 0.216i)16-s + 1.33·17-s − 0.157·19-s + (0.186 − 0.464i)20-s + (−0.393 + 0.227i)22-s + (0.399 + 0.692i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.471 - 0.881i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ -0.471 - 0.881i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.997023368\)
\(L(\frac12)\) \(\approx\) \(1.997023368\)
\(L(2)\) 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 \( 1 + (-3.08 - 3.93i)T \)
good7 \( 1 + (-9.52 - 5.5i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-6.12 - 3.53i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (12.9 - 7.5i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 22.6T + 289T^{2} \)
19 \( 1 + 3T + 361T^{2} \)
23 \( 1 + (-9.19 - 15.9i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (11.0 + 6.36i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 65iT - 1.36e3T^{2} \)
41 \( 1 + (-67.3 + 38.8i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (27.7 + 16i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-28.2 + 48.9i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 12.7T + 2.80e3T^{2} \)
59 \( 1 + (68.5 - 39.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (47.5 - 82.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (16.4 - 9.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 4.24iT - 5.04e3T^{2} \)
73 \( 1 - 119iT - 5.32e3T^{2} \)
79 \( 1 + (49.5 - 85.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (54.4 - 94.3i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 90.5iT - 7.92e3T^{2} \)
97 \( 1 + (82.2 + 47.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.15709388668454754124442978410, −9.382586385245746688430864255183, −8.710058925316009718746693756629, −7.47959555339479992821607488973, −7.19502596299284538060355776610, −5.78422275196489425102379255971, −5.37921394485106982213635553932, −4.13906678831421275678656149242, −2.48657895706133886243009126117, −1.53585308874676891588011215824, 0.840838353249169984929513582187, 1.63205480084695009072534884056, 3.03899534155142133206873701841, 4.48915436326536756722395513590, 4.96987020790675443373812549969, 6.18543892860276157226507191099, 7.66374020772715947151077022515, 8.017173034248972088656802473202, 9.053052749207642165124246249768, 9.841353439574417364891388716565

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