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Properties

Label 2-810-9.4-c3-0-7
Degree $2$
Conductor $810$
Sign $-0.173 - 0.984i$
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  + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (2.5 − 4.33i)5-s + (2 + 3.46i)7-s + 7.99·8-s − 10·10-s + (24 + 41.5i)11-s + (−1 + 1.73i)13-s + (3.99 − 6.92i)14-s + (−8 − 13.8i)16-s − 114·17-s + 140·19-s + (10 + 17.3i)20-s + (48 − 83.1i)22-s + (−36 + 62.3i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (0.107 + 0.187i)7-s + 0.353·8-s − 0.316·10-s + (0.657 + 1.13i)11-s + (−0.0213 + 0.0369i)13-s + (0.0763 − 0.132i)14-s + (−0.125 − 0.216i)16-s − 1.62·17-s + 1.69·19-s + (0.111 + 0.193i)20-s + (0.465 − 0.805i)22-s + (−0.326 + 0.565i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(47.7915\)
Root analytic conductor: \(6.91314\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6977489726\)
\(L(\frac12)\) \(\approx\) \(0.6977489726\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 \)
5 \( 1 + (-2.5 + 4.33i)T \)
good7 \( 1 + (-2 - 3.46i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-24 - 41.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 114T + 4.91e3T^{2} \)
19 \( 1 - 140T + 6.85e3T^{2} \)
23 \( 1 + (36 - 62.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (105 + 181. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (136 - 235. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 334T + 5.06e4T^{2} \)
41 \( 1 + (-99 + 171. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-134 - 232. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (108 + 187. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 78T + 1.48e5T^{2} \)
59 \( 1 + (120 - 207. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (151 + 261. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (298 - 516. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 768T + 3.57e5T^{2} \)
73 \( 1 + 478T + 3.89e5T^{2} \)
79 \( 1 + (-320 - 554. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-174 - 301. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 210T + 7.04e5T^{2} \)
97 \( 1 + (-767 - 1.32e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.948803142664595719246401922463, −9.294011631793388166509074229030, −8.737245772324103545045651916639, −7.53731043249234003750357538343, −6.85223204447930698287103087803, −5.52004068151849288517290720147, −4.59802095882482836037123259004, −3.63328139636967747284214443138, −2.23787650472173326400753020356, −1.39266072303511398267761898096, 0.20970422872946751674271682152, 1.59847307293441694674770582672, 3.09250217730309355128884354598, 4.21461360619803631157018387916, 5.42876452954204143358306610172, 6.20675495835752627143755411642, 7.04557397410445080895579418777, 7.81110752509252653158896263146, 8.984159280521740295314032792267, 9.250988954048808336708468966002

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