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Properties

Label 2-585-5.4-c3-0-61
Degree $2$
Conductor $585$
Sign $0.961 - 0.275i$
Analytic cond. $34.5161$
Root an. cond. $5.87504$
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  + 4.60i·2-s − 13.2·4-s + (3.07 + 10.7i)5-s − 15.5i·7-s − 24.1i·8-s + (−49.5 + 14.1i)10-s − 7.26·11-s − 13i·13-s + 71.4·14-s + 5.22·16-s − 27.0i·17-s − 73.2·19-s + (−40.7 − 142. i)20-s − 33.4i·22-s − 198. i·23-s + ⋯
L(s)  = 1  + 1.62i·2-s − 1.65·4-s + (0.275 + 0.961i)5-s − 0.837i·7-s − 1.06i·8-s + (−1.56 + 0.448i)10-s − 0.199·11-s − 0.277i·13-s + 1.36·14-s + 0.0815·16-s − 0.385i·17-s − 0.884·19-s + (−0.455 − 1.58i)20-s − 0.324i·22-s − 1.79i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.961 - 0.275i$
Analytic conductor: \(34.5161\)
Root analytic conductor: \(5.87504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :3/2),\ 0.961 - 0.275i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9594038348\)
\(L(\frac12)\) \(\approx\) \(0.9594038348\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.07 - 10.7i)T \)
13 \( 1 + 13iT \)
good2 \( 1 - 4.60iT - 8T^{2} \)
7 \( 1 + 15.5iT - 343T^{2} \)
11 \( 1 + 7.26T + 1.33e3T^{2} \)
17 \( 1 + 27.0iT - 4.91e3T^{2} \)
19 \( 1 + 73.2T + 6.85e3T^{2} \)
23 \( 1 + 198. iT - 1.21e4T^{2} \)
29 \( 1 - 210.T + 2.43e4T^{2} \)
31 \( 1 + 4.60T + 2.97e4T^{2} \)
37 \( 1 + 311. iT - 5.06e4T^{2} \)
41 \( 1 + 266.T + 6.89e4T^{2} \)
43 \( 1 + 350. iT - 7.95e4T^{2} \)
47 \( 1 - 89.0iT - 1.03e5T^{2} \)
53 \( 1 - 409. iT - 1.48e5T^{2} \)
59 \( 1 + 69.1T + 2.05e5T^{2} \)
61 \( 1 + 705.T + 2.26e5T^{2} \)
67 \( 1 + 285. iT - 3.00e5T^{2} \)
71 \( 1 - 728.T + 3.57e5T^{2} \)
73 \( 1 + 42.7iT - 3.89e5T^{2} \)
79 \( 1 - 486.T + 4.93e5T^{2} \)
83 \( 1 - 259. iT - 5.71e5T^{2} \)
89 \( 1 + 71.9T + 7.04e5T^{2} \)
97 \( 1 + 953. iT - 9.12e5T^{2} \)
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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

−10.44401362341546030967864526695, −9.184947954203871274838781089959, −8.267029503272124003766265718250, −7.43972525620326052752600686094, −6.71958796144071236774963397030, −6.15265580748379146171269272848, −4.98119071231980371629322829918, −4.00234558067079821274801266775, −2.50957691594995586459540861569, −0.29599492760601881274351515087, 1.25010919208286693002763423347, 2.11387715444861443314239002249, 3.27499055214349735437173039131, 4.45120881533230565149112704349, 5.25824382641027597974262444968, 6.41848143280733160913819759528, 8.134488716159908789755351516684, 8.807596116668484309643064860689, 9.589767208905261861234458493424, 10.21895315863250728190147231193

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