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Properties

Label 2-810-9.7-c3-0-10
Degree $2$
Conductor $810$
Sign $-0.766 - 0.642i$
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 + (−21 + 36.3i)11-s + (−10 − 17.3i)13-s + (3.99 + 6.92i)14-s + (−8 + 13.8i)16-s + 93·17-s + 59·19-s + (10 − 17.3i)20-s + (−42 − 72.7i)22-s + (−4.5 − 7.79i)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 + (−0.575 + 0.996i)11-s + (−0.213 − 0.369i)13-s + (0.0763 + 0.132i)14-s + (−0.125 + 0.216i)16-s + 1.32·17-s + 0.712·19-s + (0.111 − 0.193i)20-s + (−0.407 − 0.704i)22-s + (−0.0407 − 0.0706i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(47.7915\)
Root analytic conductor: \(6.91314\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :3/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.241698578\)
\(L(\frac12)\) \(\approx\) \(1.241698578\)
\(L(\frac{5}{2})\) 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 \( 1 + (21 - 36.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (10 + 17.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 93T + 4.91e3T^{2} \)
19 \( 1 - 59T + 6.85e3T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (60 - 103. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (23.5 + 40.7i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 262T + 5.06e4T^{2} \)
41 \( 1 + (63 + 109. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-89 + 154. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (72 - 124. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 741T + 1.48e5T^{2} \)
59 \( 1 + (-222 - 384. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (110.5 - 191. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-269 - 465. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 690T + 3.57e5T^{2} \)
73 \( 1 + 1.12e3T + 3.89e5T^{2} \)
79 \( 1 + (332.5 - 575. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (37.5 - 64.9i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.08e3T + 7.04e5T^{2} \)
97 \( 1 + (772 - 1.33e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.15964445527995928547063944557, −9.412087943047277647548016605670, −8.369767862771580510344024909728, −7.37467071759315143195909536313, −7.11709146750802449144862451630, −5.69291646384217765238515103626, −5.18778693460000257478791995435, −3.86112957081186773296972175141, −2.56800762964511819174453081283, −1.19479600255922974365978127340, 0.41053143475731340760894048316, 1.59018223695680602645219575109, 2.85382063271838268532206752622, 3.78391039377971092125854176048, 5.13374816996090467860421100009, 5.76520253857933826285823931030, 7.13788510808669344188603481550, 8.076580631829488279265429604700, 8.705959307772179576823195293378, 9.668804485263512851769782233707

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