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Properties

Label 2-405-9.4-c3-0-5
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  + (2.5 + 4.33i)2-s + (−8.50 + 14.7i)4-s + (−2.5 + 4.33i)5-s + (−4.5 − 7.79i)7-s − 45.0·8-s − 25.0·10-s + (4 + 6.92i)11-s + (−21.5 + 37.2i)13-s + (22.5 − 38.9i)14-s + (−44.5 − 77.0i)16-s − 122·17-s − 59·19-s + (−42.5 − 73.6i)20-s + (−20 + 34.6i)22-s + (106.5 − 184. i)23-s + ⋯
L(s)  = 1  + (0.883 + 1.53i)2-s + (−1.06 + 1.84i)4-s + (−0.223 + 0.387i)5-s + (−0.242 − 0.420i)7-s − 1.98·8-s − 0.790·10-s + (0.109 + 0.189i)11-s + (−0.458 + 0.794i)13-s + (0.429 − 0.743i)14-s + (−0.695 − 1.20i)16-s − 1.74·17-s − 0.712·19-s + (−0.475 − 0.823i)20-s + (−0.193 + 0.335i)22-s + (0.965 − 1.67i)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

\(L(2)\) \(\approx\) \(0.7849320487\)
\(L(\frac12)\) \(\approx\) \(0.7849320487\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.5 + 7.79i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-4 - 6.92i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (21.5 - 37.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 122T + 4.91e3T^{2} \)
19 \( 1 + 59T + 6.85e3T^{2} \)
23 \( 1 + (-106.5 + 184. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (112 + 193. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-18 + 31.1i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 206T + 5.06e4T^{2} \)
41 \( 1 + (206.5 - 357. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-196 - 339. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-155.5 - 269. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 377T + 1.48e5T^{2} \)
59 \( 1 + (168.5 - 291. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (20 + 34.6i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (174 - 301. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 62T + 3.57e5T^{2} \)
73 \( 1 + 1.21e3T + 3.89e5T^{2} \)
79 \( 1 + (-147 - 254. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (267 + 462. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 810T + 7.04e5T^{2} \)
97 \( 1 + (-464 - 803. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.69800744582416750952454736064, −10.75026124004093082385894414401, −9.387259108296648797379309412181, −8.474743232255682079322211580297, −7.47756531622510158647089762337, −6.65107541746026602479410518852, −6.21123123898673196640124080529, −4.48741198414787853081740684821, −4.33752899852849617482190129252, −2.63967504927610548993833761275, 0.18418789032562324421490103328, 1.72949167696162632707072665513, 2.87662664633334554027952456553, 3.90145598083978401579970537865, 4.95506520429868158166634665382, 5.74571551970640828631512640304, 7.20488945951270260611466606195, 8.794063490711800750748397891028, 9.345865679972970398183574779371, 10.55319215565430941041496125229

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