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Properties

Label 2-810-5.4-c3-0-30
Degree $2$
Conductor $810$
Sign $-0.953 + 0.301i$
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 + (3.36 + 10.6i)5-s + 27.8i·7-s − 8i·8-s + (−21.3 + 6.73i)10-s + 13.2·11-s + 54.4i·13-s − 55.6·14-s + 16·16-s + 28.7i·17-s + 109.·19-s + (−13.4 − 42.6i)20-s + 26.4i·22-s + 176. i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.301 + 0.953i)5-s + 1.50i·7-s − 0.353i·8-s + (−0.674 + 0.212i)10-s + 0.361·11-s + 1.16i·13-s − 1.06·14-s + 0.250·16-s + 0.410i·17-s + 1.32·19-s + (−0.150 − 0.476i)20-s + 0.255i·22-s + 1.60i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.953 + 0.301i$
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: \((2,\ 810,\ (\ :3/2),\ -0.953 + 0.301i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.978934584\)
\(L(\frac12)\) \(\approx\) \(1.978934584\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.36 - 10.6i)T \)
good7 \( 1 - 27.8iT - 343T^{2} \)
11 \( 1 - 13.2T + 1.33e3T^{2} \)
13 \( 1 - 54.4iT - 2.19e3T^{2} \)
17 \( 1 - 28.7iT - 4.91e3T^{2} \)
19 \( 1 - 109.T + 6.85e3T^{2} \)
23 \( 1 - 176. iT - 1.21e4T^{2} \)
29 \( 1 - 68.8T + 2.43e4T^{2} \)
31 \( 1 + 11.3T + 2.97e4T^{2} \)
37 \( 1 + 73.9iT - 5.06e4T^{2} \)
41 \( 1 - 320.T + 6.89e4T^{2} \)
43 \( 1 + 403. iT - 7.95e4T^{2} \)
47 \( 1 - 400. iT - 1.03e5T^{2} \)
53 \( 1 + 106. iT - 1.48e5T^{2} \)
59 \( 1 - 434.T + 2.05e5T^{2} \)
61 \( 1 - 812.T + 2.26e5T^{2} \)
67 \( 1 - 411. iT - 3.00e5T^{2} \)
71 \( 1 + 254.T + 3.57e5T^{2} \)
73 \( 1 + 586. iT - 3.89e5T^{2} \)
79 \( 1 + 1.35e3T + 4.93e5T^{2} \)
83 \( 1 + 752. iT - 5.71e5T^{2} \)
89 \( 1 + 1.55e3T + 7.04e5T^{2} \)
97 \( 1 + 816. iT - 9.12e5T^{2} \)
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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

−10.01039169889820317877474760966, −9.351395122536100340317854363519, −8.743009354399470988109938801260, −7.56539711865008718892543032595, −6.86293069525421676075857030622, −5.89224758558411029740466682785, −5.42699359373176791888494613033, −3.98100791023143308855501196712, −2.86220030994256645266021020238, −1.69622047092740210548819898228, 0.62532817631652801605204159546, 1.11013987304757603036125882511, 2.73857273264401053546307640316, 3.90669438020426885397198553718, 4.69359914818304780008684774519, 5.59386237614709180812440199282, 6.87681013643337836745568727158, 7.87041032742478185698218687446, 8.591820916178147046662568962231, 9.721711641415808794454368451111

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