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Properties

Label 2-810-5.4-c3-0-2
Degree $2$
Conductor $810$
Sign $0.150 - 0.988i$
Analytic cond. $47.7915$
Root an. cond. $6.91314$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 4·4-s + (−11.0 − 1.68i)5-s − 2.48i·7-s + 8i·8-s + (−3.37 + 22.1i)10-s + 15.4·11-s − 45.7i·13-s − 4.97·14-s + 16·16-s − 38.0i·17-s − 31.4·19-s + (44.2 + 6.74i)20-s − 30.8i·22-s − 165. i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.988 − 0.150i)5-s − 0.134i·7-s + 0.353i·8-s + (−0.106 + 0.699i)10-s + 0.422·11-s − 0.976i·13-s − 0.0949·14-s + 0.250·16-s − 0.542i·17-s − 0.379·19-s + (0.494 + 0.0753i)20-s − 0.298i·22-s − 1.50i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.150 - 0.988i$
Analytic conductor: \(47.7915\)
Root analytic conductor: \(6.91314\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :3/2),\ 0.150 - 0.988i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1305838176\)
\(L(\frac12)\) \(\approx\) \(0.1305838176\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 \)
5 \( 1 + (11.0 + 1.68i)T \)
good7 \( 1 + 2.48iT - 343T^{2} \)
11 \( 1 - 15.4T + 1.33e3T^{2} \)
13 \( 1 + 45.7iT - 2.19e3T^{2} \)
17 \( 1 + 38.0iT - 4.91e3T^{2} \)
19 \( 1 + 31.4T + 6.85e3T^{2} \)
23 \( 1 + 165. iT - 1.21e4T^{2} \)
29 \( 1 + 227.T + 2.43e4T^{2} \)
31 \( 1 + 50.1T + 2.97e4T^{2} \)
37 \( 1 - 150. iT - 5.06e4T^{2} \)
41 \( 1 + 180.T + 6.89e4T^{2} \)
43 \( 1 - 4.35iT - 7.95e4T^{2} \)
47 \( 1 - 384. iT - 1.03e5T^{2} \)
53 \( 1 + 166. iT - 1.48e5T^{2} \)
59 \( 1 + 488.T + 2.05e5T^{2} \)
61 \( 1 - 319.T + 2.26e5T^{2} \)
67 \( 1 - 626. iT - 3.00e5T^{2} \)
71 \( 1 + 106.T + 3.57e5T^{2} \)
73 \( 1 - 462. iT - 3.89e5T^{2} \)
79 \( 1 - 669.T + 4.93e5T^{2} \)
83 \( 1 - 1.07e3iT - 5.71e5T^{2} \)
89 \( 1 + 452.T + 7.04e5T^{2} \)
97 \( 1 - 756. iT - 9.12e5T^{2} \)
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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

−10.22670501344523173328882995317, −9.206988837829731076237286855763, −8.421632681353661654294427261077, −7.67269521102309319005896781270, −6.67928498408531860190378845182, −5.39268034895555244564259001575, −4.42393627896689855943376632320, −3.60206092790340694980629699340, −2.57579957058570686971669640851, −1.01100495591748436236019692814, 0.04197263939621606718830369502, 1.74940275312511705184438045115, 3.53199260804775884530728560320, 4.13501259368595654951993136819, 5.27572125209153937376310763533, 6.29640223644803165365248048728, 7.18712060988967599012729771549, 7.76954297969354906946422255801, 8.817232894308973671290409636883, 9.329936172863825187740831837280

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