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Properties

Label 2-585-13.4-c1-0-23
Degree $2$
Conductor $585$
Sign $-0.982 + 0.184i$
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  + (0.190 + 0.109i)2-s + (−0.975 − 1.69i)4-s − i·5-s + (−0.287 + 0.166i)7-s − 0.868i·8-s + (0.109 − 0.190i)10-s + (−4.65 − 2.68i)11-s + (−3.55 + 0.619i)13-s − 0.0729·14-s + (−1.85 + 3.21i)16-s + (2.53 + 4.38i)17-s + (−1.96 + 1.13i)19-s + (−1.69 + 0.975i)20-s + (−0.590 − 1.02i)22-s + (1.41 − 2.45i)23-s + ⋯
L(s)  = 1  + (0.134 + 0.0776i)2-s + (−0.487 − 0.845i)4-s − 0.447i·5-s + (−0.108 + 0.0627i)7-s − 0.306i·8-s + (0.0347 − 0.0601i)10-s + (−1.40 − 0.809i)11-s + (−0.985 + 0.171i)13-s − 0.0195·14-s + (−0.464 + 0.803i)16-s + (0.614 + 1.06i)17-s + (−0.450 + 0.260i)19-s + (−0.377 + 0.218i)20-s + (−0.125 − 0.217i)22-s + (0.296 − 0.512i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.982 + 0.184i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.982 + 0.184i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0467650 - 0.502768i\)
\(L(\frac12)\) \(\approx\) \(0.0467650 - 0.502768i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
13 \( 1 + (3.55 - 0.619i)T \)
good2 \( 1 + (-0.190 - 0.109i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (0.287 - 0.166i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.65 + 2.68i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.53 - 4.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.96 - 1.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.41 + 2.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.45 - 2.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.46iT - 31T^{2} \)
37 \( 1 + (5.17 + 2.98i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.23 + 1.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.53 + 4.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.34iT - 47T^{2} \)
53 \( 1 - 1.56T + 53T^{2} \)
59 \( 1 + (2.34 - 1.35i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.05 + 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.94 - 5.16i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.0 + 6.39i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.68iT - 73T^{2} \)
79 \( 1 - 4.51T + 79T^{2} \)
83 \( 1 + 4.26iT - 83T^{2} \)
89 \( 1 + (-2.79 - 1.61i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.17 + 1.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.37744007533969405295625670936, −9.468278634632558932176602446021, −8.554035635875680634612128422242, −7.77692809712983821217893710626, −6.44555459778123682397453090840, −5.49339848538152149278825905820, −4.92452497690962966766650382303, −3.63627390112111788507055272873, −2.04406286236363011702632891925, −0.25691910531959078533441559273, 2.47757698319561562827675623969, 3.25486033431590743931193426339, 4.69676115196626819266103478217, 5.27135249366027281524731356881, 6.95193918111460662873714280377, 7.53452252024009483456428867634, 8.308917310477364434543490508677, 9.547377684564257110167323281983, 10.08675035203679545905019536886, 11.17056931618683765596442733718

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