Properties

Label 2-585-65.29-c1-0-25
Degree $2$
Conductor $585$
Sign $0.241 + 0.970i$
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  + (−0.286 + 0.165i)2-s + (−0.945 + 1.63i)4-s + (2.12 − 0.702i)5-s + (−2.90 − 1.67i)7-s − 1.28i·8-s + (−0.492 + 0.552i)10-s + (−1.62 − 2.81i)11-s + (1.21 − 3.39i)13-s + 1.10·14-s + (−1.67 − 2.90i)16-s + (−1.68 − 0.974i)17-s + (−0.622 + 1.07i)19-s + (−0.856 + 4.13i)20-s + (0.929 + 0.536i)22-s + (2.33 − 1.34i)23-s + ⋯
L(s)  = 1  + (−0.202 + 0.116i)2-s + (−0.472 + 0.818i)4-s + (0.949 − 0.314i)5-s + (−1.09 − 0.633i)7-s − 0.455i·8-s + (−0.155 + 0.174i)10-s + (−0.489 − 0.847i)11-s + (0.337 − 0.941i)13-s + 0.296·14-s + (−0.419 − 0.726i)16-s + (−0.409 − 0.236i)17-s + (−0.142 + 0.247i)19-s + (−0.191 + 0.925i)20-s + (0.198 + 0.114i)22-s + (0.486 − 0.280i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.241 + 0.970i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.241 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723606 - 0.565796i\)
\(L(\frac12)\) \(\approx\) \(0.723606 - 0.565796i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 0.702i)T \)
13 \( 1 + (-1.21 + 3.39i)T \)
good2 \( 1 + (0.286 - 0.165i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (2.90 + 1.67i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.62 + 2.81i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.68 + 0.974i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.622 - 1.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.33 + 1.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.78T + 31T^{2} \)
37 \( 1 + (-1.68 + 0.974i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.39 - 2.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.56 + 4.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.86iT - 47T^{2} \)
53 \( 1 + 12.8iT - 53T^{2} \)
59 \( 1 + (-1.26 + 2.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.74 + 6.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.47 - 2.00i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.62 + 4.54i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.46iT - 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 - 8.61iT - 83T^{2} \)
89 \( 1 + (5.15 + 8.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.56 - 2.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.22283656189159909597064806031, −9.694646161133369266419422161373, −8.717226548204363869944613953712, −8.053632514760704700751871343823, −6.88472789514643115506299329799, −6.05657181805473690469247858362, −4.94947421778548327473845336755, −3.64098810825763121051290180077, −2.77250270747782686014676919063, −0.55234955533891122290873390927, 1.71674320660499278305593747918, 2.80348721626429249239269928139, 4.45855522151092345449525162755, 5.50991113248732159046930671397, 6.29710804236761438367157116692, 7.02516872823774381017813696567, 8.708387163150166616362933690752, 9.281728501453999368060022523879, 9.933841147053399691746876936514, 10.57448719938312993748340742760

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