Properties

Label 2-405-9.7-c3-0-13
Degree $2$
Conductor $405$
Sign $-0.939 + 0.342i$
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  + (−2.60 + 4.50i)2-s + (−9.52 − 16.4i)4-s + (2.5 + 4.33i)5-s + (−12.2 + 21.1i)7-s + 57.4·8-s − 26.0·10-s + (14.4 − 25.1i)11-s + (32.6 + 56.6i)13-s + (−63.4 − 109. i)14-s + (−73.2 + 126. i)16-s + 68.1·17-s + 104.·19-s + (47.6 − 82.4i)20-s + (75.3 + 130. i)22-s + (77.4 + 134. i)23-s + ⋯
L(s)  = 1  + (−0.919 + 1.59i)2-s + (−1.19 − 2.06i)4-s + (0.223 + 0.387i)5-s + (−0.658 + 1.14i)7-s + 2.53·8-s − 0.822·10-s + (0.397 − 0.688i)11-s + (0.697 + 1.20i)13-s + (−1.21 − 2.09i)14-s + (−1.14 + 1.98i)16-s + 0.972·17-s + 1.26·19-s + (0.532 − 0.922i)20-s + (0.730 + 1.26i)22-s + (0.701 + 1.21i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.939 + 0.342i$
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: \((2,\ 405,\ (\ :3/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.002089122\)
\(L(\frac12)\) \(\approx\) \(1.002089122\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.60 - 4.50i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (12.2 - 21.1i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-14.4 + 25.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-32.6 - 56.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 68.1T + 4.91e3T^{2} \)
19 \( 1 - 104.T + 6.85e3T^{2} \)
23 \( 1 + (-77.4 - 134. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-102. + 178. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-9.12 - 15.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + (-97.9 - 169. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (167. - 290. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-2.50 + 4.33i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 319.T + 1.48e5T^{2} \)
59 \( 1 + (215. + 372. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (297. - 514. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (97.9 + 169. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 425.T + 3.57e5T^{2} \)
73 \( 1 - 929.T + 3.89e5T^{2} \)
79 \( 1 + (12.2 - 21.1i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-272. + 472. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 84.1T + 7.04e5T^{2} \)
97 \( 1 + (413. - 716. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.17433988923134811903825828855, −9.750778100917915478845383864481, −9.402347463907572586629620835572, −8.584656048308783637967954689644, −7.61644660386908196227482523217, −6.55248794447728205832758598794, −6.02493158310532974860156963541, −5.15891709993940923021704398767, −3.30334322728650956723045256184, −1.29243463880616166966207908099, 0.57117780841069987616588325583, 1.35922898951436535848849845228, 3.05709744972290142118562456616, 3.74721732593318381586556115251, 5.10519808507630973869623026683, 6.89846408083559547157506040530, 7.87663347213059101267569251599, 8.831455286732166945351144675053, 9.711296167230333953018142768467, 10.37453902342126574490060844370

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