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Properties

Label 2-585-13.4-c1-0-9
Degree $2$
Conductor $585$
Sign $-0.400 - 0.916i$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.16 + 1.24i)2-s + (2.11 + 3.66i)4-s + i·5-s + (−1.64 + 0.952i)7-s + 5.55i·8-s + (−1.24 + 2.16i)10-s + (−0.926 − 0.534i)11-s + (1.40 + 3.32i)13-s − 4.75·14-s + (−2.70 + 4.69i)16-s + (−0.318 − 0.551i)17-s + (4.96 − 2.86i)19-s + (−3.66 + 2.11i)20-s + (−1.33 − 2.31i)22-s + (−1.90 + 3.30i)23-s + ⋯
L(s)  = 1  + (1.52 + 0.882i)2-s + (1.05 + 1.83i)4-s + 0.447i·5-s + (−0.623 + 0.360i)7-s + 1.96i·8-s + (−0.394 + 0.683i)10-s + (−0.279 − 0.161i)11-s + (0.388 + 0.921i)13-s − 1.27·14-s + (−0.677 + 1.17i)16-s + (−0.0772 − 0.133i)17-s + (1.13 − 0.657i)19-s + (−0.818 + 0.472i)20-s + (−0.284 − 0.492i)22-s + (−0.398 + 0.689i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.400 - 0.916i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.400 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73438 + 2.65017i\)
\(L(\frac12)\) \(\approx\) \(1.73438 + 2.65017i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-1.40 - 3.32i)T \)
good2 \( 1 + (-2.16 - 1.24i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (1.64 - 0.952i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.926 + 0.534i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.318 + 0.551i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.96 + 2.86i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.90 - 3.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.72 + 8.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.46iT - 31T^{2} \)
37 \( 1 + (-0.655 - 0.378i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.232 - 0.133i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.318 - 0.551i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.44iT - 47T^{2} \)
53 \( 1 - 6.99T + 53T^{2} \)
59 \( 1 + (-0.641 + 0.370i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.09 + 3.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.01 + 4.04i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.45 - 4.88i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.71iT - 73T^{2} \)
79 \( 1 + 9.31T + 79T^{2} \)
83 \( 1 + 5.11iT - 83T^{2} \)
89 \( 1 + (-10.8 - 6.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.65 - 2.11i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.67548799113006182448213346310, −10.12570858255325718481337185546, −9.118633654581180327459798247076, −7.900312900614562620659782804553, −7.06580407697835379605629820010, −6.29789983034507004994667374709, −5.59153709589725024933001071906, −4.49378632444673490798062651282, −3.50220205801657150572775888764, −2.55521169258056016509281208109, 1.24123091516811397613272351305, 2.83400809327471299666146453594, 3.61485854056919118572135320720, 4.68675036033394372304599527408, 5.54054013892391309704438231144, 6.36211740969228007969841117605, 7.57289528040362840088655184323, 8.820095087198857755879805967650, 10.16160028455703120647089297498, 10.43058999130246483763966676521

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