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Properties

Label 2-405-9.4-c3-0-6
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  + (2.72 + 4.72i)2-s + (−10.8 + 18.8i)4-s + (2.5 − 4.33i)5-s + (5.90 + 10.2i)7-s − 75.3·8-s + 27.2·10-s + (−28.1 − 48.7i)11-s + (−17.2 + 29.9i)13-s + (−32.2 + 55.8i)14-s + (−118. − 205. i)16-s − 39.2·17-s − 146.·19-s + (54.4 + 94.3i)20-s + (153. − 265. i)22-s + (−11.7 + 20.4i)23-s + ⋯
L(s)  = 1  + (0.964 + 1.67i)2-s + (−1.36 + 2.35i)4-s + (0.223 − 0.387i)5-s + (0.318 + 0.552i)7-s − 3.32·8-s + 0.863·10-s + (−0.770 − 1.33i)11-s + (−0.369 + 0.639i)13-s + (−0.615 + 1.06i)14-s + (−1.84 − 3.20i)16-s − 0.560·17-s − 1.76·19-s + (0.609 + 1.05i)20-s + (1.48 − 2.57i)22-s + (−0.106 + 0.185i)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: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9142265283\)
\(L(\frac12)\) \(\approx\) \(0.9142265283\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.72 - 4.72i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-5.90 - 10.2i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (28.1 + 48.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (17.2 - 29.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 39.2T + 4.91e3T^{2} \)
19 \( 1 + 146.T + 6.85e3T^{2} \)
23 \( 1 + (11.7 - 20.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-80.5 - 139. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-14.7 + 25.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 217.T + 5.06e4T^{2} \)
41 \( 1 + (-71.1 + 123. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-234. - 405. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-197. - 341. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 134.T + 1.48e5T^{2} \)
59 \( 1 + (65.5 - 113. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (129. + 224. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (222. - 385. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 560.T + 3.57e5T^{2} \)
73 \( 1 + 88.6T + 3.89e5T^{2} \)
79 \( 1 + (225. + 390. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-142. - 246. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 625.T + 7.04e5T^{2} \)
97 \( 1 + (-96.6 - 167. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.93863787614944708461176707457, −10.75957337016333461762870011356, −8.998732373358810126682504628944, −8.627527968799142941498323354537, −7.74199155865062614443142501211, −6.57078449277301893363258385739, −5.86040924338431838671865863765, −4.99987031855558922792452249623, −4.10824151821342773621331874258, −2.66356653340541148682512127676, 0.20286553645390218391544323662, 1.92925186667533687559979927281, 2.63838422845495727945446969161, 4.10573871059460405852811956766, 4.72717062766255060109147330791, 5.85187980729191149239131415119, 7.12459093286547223426266193428, 8.606736101515439971069034128603, 9.854058931291739894693732248522, 10.46650600631741181612864536676

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