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Properties

Label 2-810-45.32-c1-0-20
Degree $2$
Conductor $810$
Sign $0.993 + 0.116i$
Analytic cond. $6.46788$
Root an. cond. $2.54320$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (1.48 + 1.67i)5-s + (1.13 − 4.23i)7-s + (0.707 + 0.707i)8-s + (1 + 2i)10-s + (1.22 − 0.707i)11-s + (−1.5 − 5.59i)13-s + (2.19 − 3.79i)14-s + (0.500 + 0.866i)16-s + (4.38 − 4.38i)17-s + 3.19i·19-s + (0.448 + 2.19i)20-s + (1.36 − 0.366i)22-s + (−4.82 + 1.29i)23-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.433 + 0.249i)4-s + (0.663 + 0.748i)5-s + (0.428 − 1.59i)7-s + (0.249 + 0.249i)8-s + (0.316 + 0.632i)10-s + (0.369 − 0.213i)11-s + (−0.416 − 1.55i)13-s + (0.585 − 1.01i)14-s + (0.125 + 0.216i)16-s + (1.06 − 1.06i)17-s + 0.733i·19-s + (0.100 + 0.489i)20-s + (0.291 − 0.0780i)22-s + (−1.00 + 0.269i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.993 + 0.116i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.73396 - 0.160451i\)
\(L(\frac12)\) \(\approx\) \(2.73396 - 0.160451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-1.48 - 1.67i)T \)
good7 \( 1 + (-1.13 + 4.23i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.22 + 0.707i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.5 + 5.59i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-4.38 + 4.38i)T - 17iT^{2} \)
19 \( 1 - 3.19iT - 19T^{2} \)
23 \( 1 + (4.82 - 1.29i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.82 - 4.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.09 - 5.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.19 - 5.19i)T + 37iT^{2} \)
41 \( 1 + (-1.10 - 0.637i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.09 + 1.90i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.776 - 0.208i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.26 - 2.26i)T + 53iT^{2} \)
59 \( 1 + (-1.48 + 2.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.90 + 3.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.73 - 1.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (-4 + 4i)T - 73iT^{2} \)
79 \( 1 + (1.90 - 1.09i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.67 - 13.7i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.69T + 89T^{2} \)
97 \( 1 + (2.19 - 8.19i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.25092002112120548496790566753, −9.835066832792376573415548058059, −8.160564131771560968656705058437, −7.47953077830390740868421141489, −6.81943072342040915056363109810, −5.73864143601038929298284735871, −4.96981767440499036773132554945, −3.68291766453689388369064808400, −2.97590267053564697122951839483, −1.28477549528742574636731711102, 1.76342370584210513215646323170, 2.39196470727830129730738437731, 4.08749008622981947767824024638, 4.89068847558604093024365212196, 5.85859745162081054364196288943, 6.31131387360741842596236458312, 7.76898062026949532823054109995, 8.767337210754313782165812470197, 9.375657169057014642223748244541, 10.15022065572565177060281264294

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