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Properties

Label 2-405-45.43-c2-0-40
Degree $2$
Conductor $405$
Sign $-0.999 - 0.00266i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
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  + (−0.578 − 2.15i)2-s + (−0.866 + 0.5i)4-s + (4.89 − 1.00i)5-s + (−1.83 − 6.83i)7-s + (−4.74 − 4.74i)8-s + (−5 − 10.0i)10-s + (−7.90 + 13.6i)11-s + (3.66 − 13.6i)13-s + (−13.6 + 7.90i)14-s + (−9.49 + 16.4i)16-s + (3.16 − 3.16i)17-s − 18i·19-s + (−3.74 + 3.31i)20-s + (34.1 + 9.15i)22-s + (1.15 − 4.31i)23-s + ⋯
L(s)  = 1  + (−0.289 − 1.07i)2-s + (−0.216 + 0.125i)4-s + (0.979 − 0.200i)5-s + (−0.261 − 0.975i)7-s + (−0.592 − 0.592i)8-s + (−0.5 − 1.00i)10-s + (−0.718 + 1.24i)11-s + (0.281 − 1.05i)13-s + (−0.978 + 0.564i)14-s + (−0.593 + 1.02i)16-s + (0.186 − 0.186i)17-s − 0.947i·19-s + (−0.187 + 0.165i)20-s + (1.55 + 0.415i)22-s + (0.0503 − 0.187i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00266i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.00266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.999 - 0.00266i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.999 - 0.00266i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00185079 + 1.39142i\)
\(L(\frac12)\) \(\approx\) \(0.00185079 + 1.39142i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-4.89 + 1.00i)T \)
good2 \( 1 + (0.578 + 2.15i)T + (-3.46 + 2i)T^{2} \)
7 \( 1 + (1.83 + 6.83i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (7.90 - 13.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-3.66 + 13.6i)T + (-146. - 84.5i)T^{2} \)
17 \( 1 + (-3.16 + 3.16i)T - 289iT^{2} \)
19 \( 1 + 18iT - 361T^{2} \)
23 \( 1 + (-1.15 + 4.31i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + (41.0 + 23.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-10 + 10i)T - 1.36e3iT^{2} \)
41 \( 1 + (-15.8 - 27.3i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (13.6 - 3.66i)T + (1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-15.0 - 56.1i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (25.2 + 25.2i)T + 2.80e3iT^{2} \)
59 \( 1 + (41.0 - 23.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-29 + 50.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (95.6 + 25.6i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 63.2T + 5.04e3T^{2} \)
73 \( 1 + (-55 - 55i)T + 5.32e3iT^{2} \)
79 \( 1 + (-10.3 - 6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-73.4 + 19.6i)T + (5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (1.83 + 6.83i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−10.56825604492436360903817127845, −9.800113668349454794141553387238, −9.349638864845290767118004740555, −7.83140248468933311229278226918, −6.84147411457202544339240173380, −5.71487241486456239995530710704, −4.46968934274527551023347130824, −3.04842711188112782948636152664, −2.00053678896078430869242214178, −0.62725646656248607109211676289, 2.05536650468224523762537964178, 3.28799782025716302389956245667, 5.39830571103905346020879529734, 5.84463633299566245071485931223, 6.63993391073045841483380632013, 7.77287222607639691664610541179, 8.822223606069609359750475273618, 9.214963629762623952012429592275, 10.50146890116556328365663550789, 11.43037919306720024692523330034

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