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Properties

Label 2-810-135.122-c1-0-5
Degree $2$
Conductor $810$
Sign $0.765 - 0.643i$
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  + (−0.819 − 0.573i)2-s + (0.342 + 0.939i)4-s + (2.17 − 0.533i)5-s + (1.71 + 3.68i)7-s + (0.258 − 0.965i)8-s + (−2.08 − 0.808i)10-s + (0.396 + 0.472i)11-s + (2.48 + 3.55i)13-s + (0.705 − 3.99i)14-s + (−0.766 + 0.642i)16-s + (0.235 + 0.880i)17-s + (−5.18 + 2.99i)19-s + (1.24 + 1.85i)20-s + (−0.0537 − 0.613i)22-s + (−4.35 − 2.03i)23-s + ⋯
L(s)  = 1  + (−0.579 − 0.405i)2-s + (0.171 + 0.469i)4-s + (0.971 − 0.238i)5-s + (0.648 + 1.39i)7-s + (0.0915 − 0.341i)8-s + (−0.659 − 0.255i)10-s + (0.119 + 0.142i)11-s + (0.690 + 0.985i)13-s + (0.188 − 1.06i)14-s + (−0.191 + 0.160i)16-s + (0.0572 + 0.213i)17-s + (−1.18 + 0.686i)19-s + (0.278 + 0.415i)20-s + (−0.0114 − 0.130i)22-s + (−0.908 − 0.423i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.765 - 0.643i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.765 - 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33021 + 0.484650i\)
\(L(\frac12)\) \(\approx\) \(1.33021 + 0.484650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.819 + 0.573i)T \)
3 \( 1 \)
5 \( 1 + (-2.17 + 0.533i)T \)
good7 \( 1 + (-1.71 - 3.68i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (-0.396 - 0.472i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-2.48 - 3.55i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.235 - 0.880i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.18 - 2.99i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.35 + 2.03i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-0.993 - 5.63i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (6.86 - 2.49i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-2.63 + 0.705i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.18 + 0.385i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.847 + 9.68i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (0.383 - 0.179i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (5.83 + 5.83i)T + 53iT^{2} \)
59 \( 1 + (-6.49 - 5.44i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-11.5 - 4.19i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.78 + 1.95i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-0.354 - 0.204i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.20 - 1.93i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-11.9 + 2.11i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-8.59 + 12.2i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (0.995 + 1.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-18.9 - 1.65i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.33942376659532523938344084115, −9.334790932200728924901437955428, −8.745458102655473834620107516480, −8.270803434275079144239468259480, −6.79024167405237823458692323981, −5.99060480067010123116558815954, −5.10311512531546576603023783830, −3.82988502932023092509292641179, −2.20190267160678253402611385506, −1.75255824352371159806752903824, 0.884228511591774001621006442081, 2.17824549798002456884471750432, 3.75687246343866563505946970040, 4.92347408457906035496814327468, 5.99037463770018565052575684443, 6.68131053102238474713639743750, 7.71363128217169588824259586834, 8.286901541158318723578207898583, 9.441409024888206663116328561747, 10.09385233947839470621478288875

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