Properties

Label 2-405-5.4-c3-0-4
Degree $2$
Conductor $405$
Sign $0.514 + 0.857i$
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  + 5.02i·2-s − 17.2·4-s + (5.75 + 9.58i)5-s + 5.38i·7-s − 46.4i·8-s + (−48.1 + 28.9i)10-s − 39.1·11-s + 86.6i·13-s − 27.0·14-s + 95.2·16-s − 15.4i·17-s + 26.8·19-s + (−99.2 − 165. i)20-s − 196. i·22-s − 111. i·23-s + ⋯
L(s)  = 1  + 1.77i·2-s − 2.15·4-s + (0.514 + 0.857i)5-s + 0.290i·7-s − 2.05i·8-s + (−1.52 + 0.914i)10-s − 1.07·11-s + 1.84i·13-s − 0.516·14-s + 1.48·16-s − 0.219i·17-s + 0.324·19-s + (−1.10 − 1.84i)20-s − 1.90i·22-s − 1.00i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 0.514 + 0.857i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7498886702\)
\(L(\frac12)\) \(\approx\) \(0.7498886702\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.75 - 9.58i)T \)
good2 \( 1 - 5.02iT - 8T^{2} \)
7 \( 1 - 5.38iT - 343T^{2} \)
11 \( 1 + 39.1T + 1.33e3T^{2} \)
13 \( 1 - 86.6iT - 2.19e3T^{2} \)
17 \( 1 + 15.4iT - 4.91e3T^{2} \)
19 \( 1 - 26.8T + 6.85e3T^{2} \)
23 \( 1 + 111. iT - 1.21e4T^{2} \)
29 \( 1 + 49.2T + 2.43e4T^{2} \)
31 \( 1 + 179.T + 2.97e4T^{2} \)
37 \( 1 + 293. iT - 5.06e4T^{2} \)
41 \( 1 - 27.6T + 6.89e4T^{2} \)
43 \( 1 - 60.5iT - 7.95e4T^{2} \)
47 \( 1 + 96.2iT - 1.03e5T^{2} \)
53 \( 1 - 251. iT - 1.48e5T^{2} \)
59 \( 1 - 76.8T + 2.05e5T^{2} \)
61 \( 1 - 490.T + 2.26e5T^{2} \)
67 \( 1 - 238. iT - 3.00e5T^{2} \)
71 \( 1 + 640.T + 3.57e5T^{2} \)
73 \( 1 - 769. iT - 3.89e5T^{2} \)
79 \( 1 + 662.T + 4.93e5T^{2} \)
83 \( 1 + 1.30e3iT - 5.71e5T^{2} \)
89 \( 1 - 995.T + 7.04e5T^{2} \)
97 \( 1 - 814. iT - 9.12e5T^{2} \)
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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

−11.52305966745059981382894212167, −10.44132743244250180183494851289, −9.379635042642571297164277783922, −8.759213770184743931311719689229, −7.50576339320867024645822128805, −6.97091591037464652460770483757, −6.05398777083983815989199863321, −5.25491269008813661179134616521, −4.07934119383377232829384990218, −2.33080982728629619186206370344, 0.25719845132464015890693293509, 1.36523031181053002959664472489, 2.64925779315085412040338174255, 3.66791461880248161969063831065, 5.01727188008601286386319104538, 5.59750488990947118901312662809, 7.70372346844128046540809593287, 8.530237586239383812443403269653, 9.576988192832384798117961868470, 10.22771345100632476276197958922

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