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Properties

Label 2-405-9.4-c3-0-26
Degree $2$
Conductor $405$
Sign $-0.984 + 0.173i$
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  + (−2.26 − 3.92i)2-s + (−6.28 + 10.8i)4-s + (−2.5 + 4.33i)5-s + (1.31 + 2.28i)7-s + 20.6·8-s + 22.6·10-s + (−10.4 − 18.1i)11-s + (−30.4 + 52.7i)13-s + (5.97 − 10.3i)14-s + (3.35 + 5.80i)16-s + 86.8·17-s + 41.8·19-s + (−31.4 − 54.3i)20-s + (−47.4 + 82.1i)22-s + (48.6 − 84.2i)23-s + ⋯
L(s)  = 1  + (−0.801 − 1.38i)2-s + (−0.785 + 1.35i)4-s + (−0.223 + 0.387i)5-s + (0.0711 + 0.123i)7-s + 0.914·8-s + 0.716·10-s + (−0.286 − 0.496i)11-s + (−0.650 + 1.12i)13-s + (0.114 − 0.197i)14-s + (0.0523 + 0.0907i)16-s + 1.23·17-s + 0.504·19-s + (−0.351 − 0.608i)20-s + (−0.459 + 0.796i)22-s + (0.441 − 0.764i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.984 + 0.173i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -0.984 + 0.173i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6571417959\)
\(L(\frac12)\) \(\approx\) \(0.6571417959\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.26 + 3.92i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-1.31 - 2.28i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (10.4 + 18.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (30.4 - 52.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 86.8T + 4.91e3T^{2} \)
19 \( 1 - 41.8T + 6.85e3T^{2} \)
23 \( 1 + (-48.6 + 84.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (78.5 + 136. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (47.6 - 82.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 160.T + 5.06e4T^{2} \)
41 \( 1 + (-116. + 201. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (243. + 422. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-12.1 - 21.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 709.T + 1.48e5T^{2} \)
59 \( 1 + (95.9 - 166. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-372. - 644. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-411. + 713. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.06e3T + 3.57e5T^{2} \)
73 \( 1 + 132.T + 3.89e5T^{2} \)
79 \( 1 + (-352. - 610. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (709. + 1.22e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 401.T + 7.04e5T^{2} \)
97 \( 1 + (265. + 459. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.42885389245014203750084453575, −9.692103681713704241807027455023, −8.854685040755276546819933604520, −7.935873267389707169006931865239, −6.87685060082319988391454171719, −5.40788557269930285561782128063, −3.92591750900231952558700408127, −2.94255531208329593985729413811, −1.82957331246775267430878284376, −0.34501334448093173358856501434, 1.08051972773218010881653839914, 3.22726825467382469934992497052, 5.01982818628227468896391959958, 5.54350279600172102228175524590, 6.84283790681088581653877252132, 7.80162026033694463311130894654, 8.038388827650003383820883536350, 9.461305073092689207189158986645, 9.808053597401700390976441374721, 11.02679689460695515464222804038

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