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Properties

Label 2-585-585.59-c1-0-42
Degree $2$
Conductor $585$
Sign $-0.997 - 0.0682i$
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  + (−1.62 − 1.62i)2-s + (−1.49 + 0.866i)3-s + 3.26i·4-s + (0.420 − 2.19i)5-s + (3.83 + 1.02i)6-s + (−0.840 + 0.225i)7-s + (2.05 − 2.05i)8-s + (1.49 − 2.59i)9-s + (−4.24 + 2.88i)10-s + (2.47 + 2.47i)11-s + (−2.82 − 4.89i)12-s + (−3.59 − 0.286i)13-s + (1.72 + 0.998i)14-s + (1.27 + 3.65i)15-s − 0.126·16-s + (4.50 − 2.60i)17-s + ⋯
L(s)  = 1  + (−1.14 − 1.14i)2-s + (−0.866 + 0.500i)3-s + 1.63i·4-s + (0.187 − 0.982i)5-s + (1.56 + 0.419i)6-s + (−0.317 + 0.0851i)7-s + (0.725 − 0.725i)8-s + (0.499 − 0.866i)9-s + (−1.34 + 0.911i)10-s + (0.745 + 0.745i)11-s + (−0.816 − 1.41i)12-s + (−0.996 − 0.0795i)13-s + (0.462 + 0.266i)14-s + (0.328 + 0.944i)15-s − 0.0317·16-s + (1.09 − 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.997 - 0.0682i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.997 - 0.0682i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0117973 + 0.345231i\)
\(L(\frac12)\) \(\approx\) \(0.0117973 + 0.345231i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.49 - 0.866i)T \)
5 \( 1 + (-0.420 + 2.19i)T \)
13 \( 1 + (3.59 + 0.286i)T \)
good2 \( 1 + (1.62 + 1.62i)T + 2iT^{2} \)
7 \( 1 + (0.840 - 0.225i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.47 - 2.47i)T + 11iT^{2} \)
17 \( 1 + (-4.50 + 2.60i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.619 + 2.31i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.18 + 2.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.32iT - 29T^{2} \)
31 \( 1 + (7.29 - 1.95i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-4.25 + 1.14i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (8.36 + 2.24i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.59 - 4.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.49 + 0.937i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 3.57T + 53T^{2} \)
59 \( 1 + (7.30 + 7.30i)T + 59iT^{2} \)
61 \( 1 + (-5.41 + 9.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.39 + 1.98i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-8.93 - 2.39i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.0879 - 0.0879i)T + 73iT^{2} \)
79 \( 1 + (-0.0600 - 0.103i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.94 + 10.9i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (16.5 - 4.44i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.46 + 9.19i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.959693708483662912701367624570, −9.613941074644276342703211829413, −9.093402780370652898218084787946, −7.86745384023615610329281806640, −6.78678380261727576707405600588, −5.38673635379395567645142488017, −4.56003216254481492190473429979, −3.21769079268695516567714344897, −1.63458762876360977131306103968, −0.34503062892704338613413944669, 1.43142469561334659246599596161, 3.40433233418198040314288554998, 5.36053238739459767103717600735, 6.03452671552486388915262264571, 6.86079564594797352681701817094, 7.35710691709483485191895314864, 8.240741978974871327487473893615, 9.468221829638358491996698090087, 10.09643189790255032958492406759, 10.87637996239642882146671932166

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