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Properties

Label 2-405-45.43-c2-0-17
Degree $2$
Conductor $405$
Sign $-0.970 - 0.242i$
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  + (0.814 + 3.03i)2-s + (−5.10 + 2.94i)4-s + (4.99 + 0.203i)5-s + (−0.530 − 1.98i)7-s + (−4.22 − 4.22i)8-s + (3.44 + 15.3i)10-s + (−1.67 + 2.89i)11-s + (−3.82 + 14.2i)13-s + (5.58 − 3.22i)14-s + (−2.39 + 4.15i)16-s + (2.65 − 2.65i)17-s + 20.6i·19-s + (−26.1 + 13.6i)20-s + (−10.1 − 2.72i)22-s + (−6.02 + 22.4i)23-s + ⋯
L(s)  = 1  + (0.407 + 1.51i)2-s + (−1.27 + 0.737i)4-s + (0.999 + 0.0407i)5-s + (−0.0757 − 0.282i)7-s + (−0.528 − 0.528i)8-s + (0.344 + 1.53i)10-s + (−0.152 + 0.263i)11-s + (−0.294 + 1.09i)13-s + (0.398 − 0.230i)14-s + (−0.149 + 0.259i)16-s + (0.155 − 0.155i)17-s + 1.08i·19-s + (−1.30 + 0.684i)20-s + (−0.462 − 0.123i)22-s + (−0.261 + 0.976i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.970 - 0.242i$
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: \((2,\ 405,\ (\ :1),\ -0.970 - 0.242i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.268929 + 2.18222i\)
\(L(\frac12)\) \(\approx\) \(0.268929 + 2.18222i\)
\(L(2)\) 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 \( 1 + (-4.99 - 0.203i)T \)
good2 \( 1 + (-0.814 - 3.03i)T + (-3.46 + 2i)T^{2} \)
7 \( 1 + (0.530 + 1.98i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (1.67 - 2.89i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (3.82 - 14.2i)T + (-146. - 84.5i)T^{2} \)
17 \( 1 + (-2.65 + 2.65i)T - 289iT^{2} \)
19 \( 1 - 20.6iT - 361T^{2} \)
23 \( 1 + (6.02 - 22.4i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + (0.739 + 0.426i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-9.34 - 16.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-38.0 + 38.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (14.3 + 24.8i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (30.7 - 8.23i)T + (1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (7.22 + 26.9i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (28.6 + 28.6i)T + 2.80e3iT^{2} \)
59 \( 1 + (-96.9 + 55.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (47.0 - 81.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-74.9 - 20.0i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 68T + 5.04e3T^{2} \)
73 \( 1 + (39.7 + 39.7i)T + 5.32e3iT^{2} \)
79 \( 1 + (-21.2 - 12.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (28.8 - 7.74i)T + (5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 94.1iT - 7.92e3T^{2} \)
97 \( 1 + (-5.34 - 19.9i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−11.55443167581191156100957178425, −10.25586750126718476624482333938, −9.482875587614932449179976846372, −8.508817212109381108716646277184, −7.42935190374055012897395200988, −6.71509613539810709982707286958, −5.82323910660206085847444048389, −5.03033613481450107076513988576, −3.86180019919316592084804406481, −1.93135257253335742553984510663, 0.858511822619242918252071542282, 2.35951062204133709497912795114, 3.04481382670355567295657267245, 4.56643663577407663997890965579, 5.45240026775544287067377745088, 6.57650700442521060576394845622, 8.129985270665989162158887443563, 9.256806915532517760931842295872, 9.988609954564931564706307252232, 10.64535546102192264776396716721

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