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Properties

Label 2-585-65.64-c1-0-25
Degree $2$
Conductor $585$
Sign $0.992 + 0.124i$
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  + 2·2-s + 2·4-s + (2 − i)5-s + 3·7-s + (4 − 2i)10-s + 5i·11-s + (−3 − 2i)13-s + 6·14-s − 4·16-s − 3i·17-s − 6i·19-s + (4 − 2i)20-s + 10i·22-s + 9i·23-s + (3 − 4i)25-s + (−6 − 4i)26-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + (0.894 − 0.447i)5-s + 1.13·7-s + (1.26 − 0.632i)10-s + 1.50i·11-s + (−0.832 − 0.554i)13-s + 1.60·14-s − 16-s − 0.727i·17-s − 1.37i·19-s + (0.894 − 0.447i)20-s + 2.13i·22-s + 1.87i·23-s + (0.600 − 0.800i)25-s + (−1.17 − 0.784i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.992 + 0.124i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.992 + 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.49680 - 0.217703i\)
\(L(\frac12)\) \(\approx\) \(3.49680 - 0.217703i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
13 \( 1 + (3 + 2i)T \)
good2 \( 1 - 2T + 2T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 - 5iT - 11T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 - 9iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + 5iT - 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 5iT - 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 15T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - iT - 89T^{2} \)
97 \( 1 - 3T + 97T^{2} \)
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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

−10.97844215882408177988573474519, −9.704887484850753901636210390500, −9.194915851832507433526147382543, −7.72419418467460300796419528806, −6.94627373863877983068362298647, −5.66842890621948180439326955837, −4.88271407134164340871263453969, −4.59390406039760231689655375873, −2.85497385576384006808620357979, −1.81519697457318625967382379853, 1.90306447948722643718359930552, 3.02844831692419744500769815553, 4.21242420726856230888366215458, 5.16056740807084470282749955218, 5.98287322860284498591810825816, 6.61900283515020463328930676205, 8.045242613288293194955200531335, 8.856831722026921941714036657255, 10.14089860039039578414241450246, 10.93920089835723928221966530646

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