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Properties

Label 2-405-9.7-c3-0-45
Degree $2$
Conductor $405$
Sign $-0.766 - 0.642i$
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  + (2.5 − 4.33i)2-s + (−8.50 − 14.7i)4-s + (−2.5 − 4.33i)5-s + (15 − 25.9i)7-s − 45.0·8-s − 25.0·10-s + (25 − 43.3i)11-s + (10 + 17.3i)13-s + (−75.0 − 129. i)14-s + (−44.5 + 77.0i)16-s + 10·17-s − 44·19-s + (−42.5 + 73.6i)20-s + (−125 − 216. i)22-s + (60 + 103. i)23-s + ⋯
L(s)  = 1  + (0.883 − 1.53i)2-s + (−1.06 − 1.84i)4-s + (−0.223 − 0.387i)5-s + (0.809 − 1.40i)7-s − 1.98·8-s − 0.790·10-s + (0.685 − 1.18i)11-s + (0.213 + 0.369i)13-s + (−1.43 − 2.47i)14-s + (−0.695 + 1.20i)16-s + 0.142·17-s − 0.531·19-s + (−0.475 + 0.823i)20-s + (−1.21 − 2.09i)22-s + (0.543 + 0.942i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.766 - 0.642i$
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: \((2,\ 405,\ (\ :3/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.896635723\)
\(L(\frac12)\) \(\approx\) \(2.896635723\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 + 4.33i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-15 + 25.9i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-25 + 43.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-10 - 17.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 10T + 4.91e3T^{2} \)
19 \( 1 + 44T + 6.85e3T^{2} \)
23 \( 1 + (-60 - 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (25 - 43.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (54 + 93.5i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 40T + 5.06e4T^{2} \)
41 \( 1 + (-200 - 346. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (140 - 242. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (140 - 242. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 610T + 1.48e5T^{2} \)
59 \( 1 + (-25 - 43.3i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-259 + 448. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-90 - 155. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 700T + 3.57e5T^{2} \)
73 \( 1 + 410T + 3.89e5T^{2} \)
79 \( 1 + (-258 + 446. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-330 + 571. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.50e3T + 7.04e5T^{2} \)
97 \( 1 + (-815 + 1.41e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.73674653544040004898615749892, −9.712475931479315333238813080643, −8.706262883731577315153167417337, −7.53169589096493035022991262688, −6.08544944814517231495242485933, −4.85905304328510838075582007155, −4.08291056000224760857308616581, −3.31269490309500802344880968338, −1.57133919639441978373780902946, −0.77874048376382023360994366938, 2.27179706799063025876196204964, 3.84825058831471065364550313732, 4.87282790617151212743784038035, 5.64138901843282852954837576542, 6.64714773366738634578107916293, 7.40885321315159823090041990850, 8.458067917910048444916136473812, 9.002437250714761817255421035697, 10.51756728298634723215758291633, 11.88321690357005489384247346627

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