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Properties

Label 2-585-9.4-c1-0-23
Degree $2$
Conductor $585$
Sign $-0.442 + 0.896i$
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  + (−1.20 − 2.08i)2-s + (1.64 + 0.539i)3-s + (−1.90 + 3.29i)4-s + (−0.5 + 0.866i)5-s + (−0.857 − 4.08i)6-s + (−0.497 − 0.861i)7-s + 4.36·8-s + (2.41 + 1.77i)9-s + 2.41·10-s + (−2.77 − 4.80i)11-s + (−4.91 + 4.40i)12-s + (0.5 − 0.866i)13-s + (−1.19 + 2.07i)14-s + (−1.29 + 1.15i)15-s + (−1.44 − 2.50i)16-s + 5.33·17-s + ⋯
L(s)  = 1  + (−0.852 − 1.47i)2-s + (0.950 + 0.311i)3-s + (−0.952 + 1.64i)4-s + (−0.223 + 0.387i)5-s + (−0.350 − 1.66i)6-s + (−0.188 − 0.325i)7-s + 1.54·8-s + (0.806 + 0.591i)9-s + 0.762·10-s + (−0.837 − 1.44i)11-s + (−1.41 + 1.27i)12-s + (0.138 − 0.240i)13-s + (−0.320 + 0.555i)14-s + (−0.333 + 0.298i)15-s + (−0.361 − 0.626i)16-s + 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.442 + 0.896i$
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: \((2,\ 585,\ (\ :1/2),\ -0.442 + 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.601382 - 0.967843i\)
\(L(\frac12)\) \(\approx\) \(0.601382 - 0.967843i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.64 - 0.539i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.20 + 2.08i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.497 + 0.861i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.77 + 4.80i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.33T + 17T^{2} \)
19 \( 1 - 4.35T + 19T^{2} \)
23 \( 1 + (-2.82 + 4.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.44 + 5.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.67 + 2.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.77T + 37T^{2} \)
41 \( 1 + (-5.36 + 9.30i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.80 - 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.320 + 0.554i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.0369T + 53T^{2} \)
59 \( 1 + (6.89 - 11.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.48 + 2.56i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.83 - 8.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.39T + 71T^{2} \)
73 \( 1 + 6.18T + 73T^{2} \)
79 \( 1 + (0.938 + 1.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.80 - 10.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.37T + 89T^{2} \)
97 \( 1 + (2.93 + 5.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.45115300717786148054158542041, −9.709934156626213240832186757947, −8.874496883153990178111975207377, −8.044266724107510976079799088476, −7.50393739080898766325451460826, −5.69561537318163879969779946316, −4.06006049428782341655966030529, −3.18591074653952959789123609265, −2.62671687019353100053525633168, −0.872485513590632478588703694950, 1.43285430644078248323160212969, 3.20077611260353481770955634933, 4.81909239099172280689786674376, 5.66312851948741998936056371842, 7.02048547590028551340586503043, 7.51490475249644879344557340469, 8.081636609094221210895805524576, 9.252997929668348877653067577633, 9.450214816004397731327262608690, 10.42762123812127114577661496295

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