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Properties

Label 2-810-5.2-c2-0-8
Degree $2$
Conductor $810$
Sign $-0.991 + 0.129i$
Analytic cond. $22.0709$
Root an. cond. $4.69796$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + 2i·4-s + (1.76 + 4.67i)5-s + (−4.32 − 4.32i)7-s + (−2 + 2i)8-s + (−2.90 + 6.44i)10-s + 9.81·11-s + (−13.2 + 13.2i)13-s − 8.65i·14-s − 4·16-s + (−18.6 − 18.6i)17-s + 16.6i·19-s + (−9.35 + 3.53i)20-s + (9.81 + 9.81i)22-s + (−11.3 + 11.3i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (0.353 + 0.935i)5-s + (−0.618 − 0.618i)7-s + (−0.250 + 0.250i)8-s + (−0.290 + 0.644i)10-s + 0.892·11-s + (−1.02 + 1.02i)13-s − 0.618i·14-s − 0.250·16-s + (−1.09 − 1.09i)17-s + 0.874i·19-s + (−0.467 + 0.176i)20-s + (0.446 + 0.446i)22-s + (−0.494 + 0.494i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.991 + 0.129i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ -0.991 + 0.129i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.192422990\)
\(L(\frac12)\) \(\approx\) \(1.192422990\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
5 \( 1 + (-1.76 - 4.67i)T \)
good7 \( 1 + (4.32 + 4.32i)T + 49iT^{2} \)
11 \( 1 - 9.81T + 121T^{2} \)
13 \( 1 + (13.2 - 13.2i)T - 169iT^{2} \)
17 \( 1 + (18.6 + 18.6i)T + 289iT^{2} \)
19 \( 1 - 16.6iT - 361T^{2} \)
23 \( 1 + (11.3 - 11.3i)T - 529iT^{2} \)
29 \( 1 - 34.0iT - 841T^{2} \)
31 \( 1 - 29.4T + 961T^{2} \)
37 \( 1 + (27.3 + 27.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 22.4T + 1.68e3T^{2} \)
43 \( 1 + (25.2 - 25.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (44.0 + 44.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (14.0 - 14.0i)T - 2.80e3iT^{2} \)
59 \( 1 - 21.0iT - 3.48e3T^{2} \)
61 \( 1 - 69.8T + 3.72e3T^{2} \)
67 \( 1 + (-1.75 - 1.75i)T + 4.48e3iT^{2} \)
71 \( 1 + 99.0T + 5.04e3T^{2} \)
73 \( 1 + (0.175 - 0.175i)T - 5.32e3iT^{2} \)
79 \( 1 - 64.2iT - 6.24e3T^{2} \)
83 \( 1 + (-45.7 + 45.7i)T - 6.88e3iT^{2} \)
89 \( 1 + 43.0iT - 7.92e3T^{2} \)
97 \( 1 + (-117. - 117. i)T + 9.40e3iT^{2} \)
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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

−10.35383788158644712076904212028, −9.678180228883228470938124083473, −8.885181262848811437155292017333, −7.51929624493635081249347779824, −6.77481160355782433344073350588, −6.52146857411128269967826054780, −5.18578157758740541554112604632, −4.11714524500931477916982023287, −3.24167610444779028768545573262, −1.99359152335224095443421569413, 0.31110135000304958156300079324, 1.86178393500837228239851917661, 2.89646210616191885142745073559, 4.22276121089570085605486306773, 4.97687145102596684760020827132, 6.01323302007526082482147177929, 6.63994546963841371848370365390, 8.170937296567747360811053706655, 8.885654753380839217564527884335, 9.731100943882475672822978906124

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