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Properties

Label 2-585-5.4-c1-0-3
Degree $2$
Conductor $585$
Sign $0.662 + 0.749i$
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  − 2.67i·2-s − 5.15·4-s + (−1.67 + 1.48i)5-s + 0.806i·7-s + 8.44i·8-s + (3.96 + 4.48i)10-s + 3.67·11-s + i·13-s + 2.15·14-s + 12.2·16-s + 1.35i·17-s + 1.67·19-s + (8.63 − 7.63i)20-s − 9.83i·22-s + 6.48i·23-s + ⋯
L(s)  = 1  − 1.89i·2-s − 2.57·4-s + (−0.749 + 0.662i)5-s + 0.304i·7-s + 2.98i·8-s + (1.25 + 1.41i)10-s + 1.10·11-s + 0.277i·13-s + 0.576·14-s + 3.06·16-s + 0.327i·17-s + 0.384·19-s + (1.93 − 1.70i)20-s − 2.09i·22-s + 1.35i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.882300 - 0.397595i\)
\(L(\frac12)\) \(\approx\) \(0.882300 - 0.397595i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 1.48i)T \)
13 \( 1 - iT \)
good2 \( 1 + 2.67iT - 2T^{2} \)
7 \( 1 - 0.806iT - 7T^{2} \)
11 \( 1 - 3.67T + 11T^{2} \)
17 \( 1 - 1.35iT - 17T^{2} \)
19 \( 1 - 1.67T + 19T^{2} \)
23 \( 1 - 6.48iT - 23T^{2} \)
29 \( 1 - 2.41T + 29T^{2} \)
31 \( 1 + 5.28T + 31T^{2} \)
37 \( 1 - 3.76iT - 37T^{2} \)
41 \( 1 - 8.31T + 41T^{2} \)
43 \( 1 - 6.79iT - 43T^{2} \)
47 \( 1 + 3.19iT - 47T^{2} \)
53 \( 1 - 5.73iT - 53T^{2} \)
59 \( 1 - 5.98T + 59T^{2} \)
61 \( 1 + 1.76T + 61T^{2} \)
67 \( 1 - 9.89iT - 67T^{2} \)
71 \( 1 + 8.56T + 71T^{2} \)
73 \( 1 - 11.7iT - 73T^{2} \)
79 \( 1 - 2.26T + 79T^{2} \)
83 \( 1 - 3.84iT - 83T^{2} \)
89 \( 1 - 2.77T + 89T^{2} \)
97 \( 1 + 1.87iT - 97T^{2} \)
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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

−10.85208683009363453562040525887, −9.867563046735222319506349214945, −9.194995284923712443848544276456, −8.321331678040065513823565223499, −7.16172058992983089819421296486, −5.72469226326141289095688224520, −4.33846474712727425608845723228, −3.67585058022720638016301710304, −2.70272030596729739694972576361, −1.33175594654473272691716627954, 0.65263460360565691165992837320, 3.75331338795445326740238017521, 4.48921942771823225494957860959, 5.41931515407399546264626998436, 6.45275597357844479665911850473, 7.26899527139791677796032576689, 7.957182183851543767084872517011, 8.895044026241910514499770671184, 9.306291515719291899092137976240, 10.62295221096288491698068153687

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