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Properties

Label 2-585-9.4-c1-0-44
Degree $2$
Conductor $585$
Sign $-0.989 - 0.143i$
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  + (−0.817 − 1.41i)2-s + (1.40 − 1.01i)3-s + (−0.335 + 0.580i)4-s + (0.5 − 0.866i)5-s + (−2.58 − 1.15i)6-s + (−1.06 − 1.84i)7-s − 2.17·8-s + (0.940 − 2.84i)9-s − 1.63·10-s + (0.0263 + 0.0455i)11-s + (0.118 + 1.15i)12-s + (0.5 − 0.866i)13-s + (−1.74 + 3.01i)14-s + (−0.177 − 1.72i)15-s + (2.44 + 4.23i)16-s − 2.48·17-s + ⋯
L(s)  = 1  + (−0.577 − 1.00i)2-s + (0.810 − 0.585i)3-s + (−0.167 + 0.290i)4-s + (0.223 − 0.387i)5-s + (−1.05 − 0.472i)6-s + (−0.402 − 0.697i)7-s − 0.768·8-s + (0.313 − 0.949i)9-s − 0.516·10-s + (0.00793 + 0.0137i)11-s + (0.0342 + 0.333i)12-s + (0.138 − 0.240i)13-s + (−0.465 + 0.805i)14-s + (−0.0457 − 0.444i)15-s + (0.611 + 1.05i)16-s − 0.601·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.989 - 0.143i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.989 - 0.143i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0909839 + 1.25960i\)
\(L(\frac12)\) \(\approx\) \(0.0909839 + 1.25960i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.40 + 1.01i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.817 + 1.41i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1.06 + 1.84i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0263 - 0.0455i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 2.48T + 17T^{2} \)
19 \( 1 - 2.13T + 19T^{2} \)
23 \( 1 + (2.46 - 4.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.24 - 2.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.08 + 7.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.11T + 37T^{2} \)
41 \( 1 + (-2.73 + 4.74i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.73 - 8.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.88 + 8.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.64T + 53T^{2} \)
59 \( 1 + (3.74 - 6.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.89 - 5.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.11 + 5.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.50T + 71T^{2} \)
73 \( 1 + 1.10T + 73T^{2} \)
79 \( 1 + (-7.80 - 13.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.244 + 0.423i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.64T + 89T^{2} \)
97 \( 1 + (3.67 + 6.35i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.996795322975517636340017169636, −9.560247308180284459206293117470, −8.696447049286355753560512019980, −7.82833509529398232957365175246, −6.78852097814854108497509489667, −5.79657792813570335557505567564, −4.08005277085453483545678933728, −3.07655464688617898193993376958, −1.95659681196968179459465901619, −0.77459067520787707870265516409, 2.41285575460481785083753865394, 3.32844117715815408255508347222, 4.75577102133660786306433731446, 6.00168904269550221293165837414, 6.74596477818781484733939579695, 7.77373216252023511951632034970, 8.549786191639792693865859970052, 9.200778059427653314500479300205, 9.887689180120064025935094765458, 10.86564020521535431111525887786

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