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Properties

Label 2-405-9.7-c3-0-16
Degree $2$
Conductor $405$
Sign $0.173 - 0.984i$
Analytic cond. $23.8957$
Root an. cond. $4.88833$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (3.5 + 6.06i)4-s + (2.5 + 4.33i)5-s + (3 − 5.19i)7-s + 15·8-s + 5·10-s + (−23.5 + 40.7i)11-s + (2.5 + 4.33i)13-s + (−3 − 5.19i)14-s + (−20.5 + 35.5i)16-s + 131·17-s − 56·19-s + (−17.5 + 30.3i)20-s + (23.5 + 40.7i)22-s + (1.5 + 2.59i)23-s + ⋯
L(s)  = 1  + (0.176 − 0.306i)2-s + (0.437 + 0.757i)4-s + (0.223 + 0.387i)5-s + (0.161 − 0.280i)7-s + 0.662·8-s + 0.158·10-s + (−0.644 + 1.11i)11-s + (0.0533 + 0.0923i)13-s + (−0.0572 − 0.0991i)14-s + (−0.320 + 0.554i)16-s + 1.86·17-s − 0.676·19-s + (−0.195 + 0.338i)20-s + (0.227 + 0.394i)22-s + (0.0135 + 0.0235i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.272161186\)
\(L(\frac12)\) \(\approx\) \(2.272161186\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 - 4.33i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-3 + 5.19i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (23.5 - 40.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 131T + 4.91e3T^{2} \)
19 \( 1 + 56T + 6.85e3T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (78.5 - 135. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (112.5 + 194. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 70T + 5.06e4T^{2} \)
41 \( 1 + (-70 - 121. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (198.5 - 343. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (173.5 - 300. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 4T + 1.48e5T^{2} \)
59 \( 1 + (-374 - 647. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-169 + 292. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (246 + 426. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 32T + 3.57e5T^{2} \)
73 \( 1 - 970T + 3.89e5T^{2} \)
79 \( 1 + (-628.5 + 1.08e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (51 - 88.3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.48e3T + 7.04e5T^{2} \)
97 \( 1 + (487 - 843. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.00305315537488317143596368968, −10.31714912434841002131519526681, −9.396002806365963110901379082504, −7.87806314303922216421800441235, −7.54490658990155895428086736956, −6.41626014943428114089570507024, −5.09433930657291656303107411384, −3.90932428186205648874900384635, −2.83484721001250786344729583483, −1.67010086897332496822657107370, 0.70532577044397818411966240529, 2.07122568382398696308808077558, 3.56071335122131363046138089821, 5.32367268989330517480770547782, 5.52266292023024773035295058494, 6.70950385483033090187267650416, 7.86450514138027266808410492180, 8.688742893077145323964999997375, 9.911162063304479401087819879755, 10.54037367682569579920004366535

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