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Properties

Label 2-405-9.4-c3-0-1
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  + (−2.61 − 4.53i)2-s + (−9.69 + 16.7i)4-s + (−2.5 + 4.33i)5-s + (−16.5 − 28.5i)7-s + 59.5·8-s + 26.1·10-s + (−3.04 − 5.26i)11-s + (32.2 − 55.8i)13-s + (−86.3 + 149. i)14-s + (−78.3 − 135. i)16-s − 76.9·17-s − 118.·19-s + (−48.4 − 83.9i)20-s + (−15.9 + 27.5i)22-s + (−38.8 + 67.2i)23-s + ⋯
L(s)  = 1  + (−0.925 − 1.60i)2-s + (−1.21 + 2.09i)4-s + (−0.223 + 0.387i)5-s + (−0.891 − 1.54i)7-s + 2.63·8-s + 0.827·10-s + (−0.0833 − 0.144i)11-s + (0.687 − 1.19i)13-s + (−1.64 + 2.85i)14-s + (−1.22 − 2.12i)16-s − 1.09·17-s − 1.43·19-s + (−0.541 − 0.938i)20-s + (−0.154 + 0.267i)22-s + (−0.351 + 0.609i)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

\(L(2)\) \(\approx\) \(0.1145611851\)
\(L(\frac12)\) \(\approx\) \(0.1145611851\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.61 + 4.53i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (16.5 + 28.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (3.04 + 5.26i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-32.2 + 55.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 76.9T + 4.91e3T^{2} \)
19 \( 1 + 118.T + 6.85e3T^{2} \)
23 \( 1 + (38.8 - 67.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (31.4 + 54.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-53.4 + 92.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 108.T + 5.06e4T^{2} \)
41 \( 1 + (71.3 - 123. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (169. + 294. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-299. - 518. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 488.T + 1.48e5T^{2} \)
59 \( 1 + (121. - 210. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-249. - 432. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-460. + 798. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 60.6T + 3.57e5T^{2} \)
73 \( 1 + 338.T + 3.89e5T^{2} \)
79 \( 1 + (-278. - 481. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-32.4 - 56.1i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 941.T + 7.04e5T^{2} \)
97 \( 1 + (-521. - 902. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.62618071158525917884677460958, −10.34322161539012132995111103223, −9.347475936847739707514818970492, −8.309399760744459148049288533180, −7.51156815031089480260505462076, −6.35519262152877699540188949589, −4.19774551670453036638476450247, −3.61724473018249549790943511883, −2.48934643509059807346059533151, −0.867579788762575429703371294669, 0.07103975294539573573797852231, 2.07450329355344618834840184595, 4.27322936763706051436148240277, 5.43191301234365108682126608537, 6.42226460235918463012981798712, 6.77654707901060326688681890787, 8.384581754342709210313625497484, 8.748812258883021072222039700557, 9.339127877153855779569769447610, 10.39474832105815434216612594059

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